30 Jul

Problematic Postcards

If you’ve come here as a result of a puzzling postcard you may have come across, welcome to Studio Infinity! We hope you’ll enjoy looking at some of the other content below as well, but here are the three posts corresponding to the problems you can find on those postcards, each of which links to a solution.

Insubordinate Integral
Smallish Sequence
Troubling Triangle

And, as I’ve mentioned here before, I invite you all to submit a problem or solution to Math Horizons Playground.

17 Sep

Woven SSD

It’s high time that S∞ got back to its core: mathematical constructions you can build. Here’s an attractive star-shaped polyhedron made with a weaving technique that I am indebted to Jürgen Richter-Gebert for introducing me to. It’s called the “small stellated dodecahedron,” and is one of the Kepler-Poinsot regular polyhedra. (Unlike the usual five Platonic solids, its edges intersect each other at points besides the twelve vertices.) We’ll call it the “SSD” for short.

30 equal-length rodsScissors
32 rubber bands

To build this, you first need 30 rods all the same length, made of some material that has a bit of give (hint: not glass rods). You can choose almost any material; here, I’ve used mostly aluminum tubes, with five contrasting brass tubes of the same diameter to highlight the first star you make. Possibilities include thin dowels, bamboo strips, pipe cleaners, and more.

You also need 32 rubber bands. They should be sized so that it’s easy enough to get them around five rods simultaneously, but then they should hold that thickness of material securely (of course you can double or triple up the bands if need be). You can use almost any type of elastic bands: ordinary office rubber bands, colored hair elastics, or rubber O-rings as I’ve used here.

Before you get started assembling the SSD, you should mark the locations that the rubber bands will go on each rod. To find these locations, first pick an amount of each rod that will extend past each vertex for security of the connections. For example, I used 18-inch rods, so I chose that the rods would extend 1/2 inch past each vertex. If you are making a larger version, you’d want to pick something longer; if you are making a smaller SSD, I wouldn’t go much below 1/2 inch because the vertex connections might start slipping apart too easily.

Subtract twice the amount you are leaving at the vertices from the length of the rods to get the distance between the vertices. In my case, that’s 18 – 2×(1/2) = 17 inches. Now multiply this value by 2/(3+√5); for 17 inches, this gives about 6.49 inches. (Hopefully I will get a chance to write a MathStream post that explains this calculation.) Finally add back in the amount you’re leaving at the vertices; this step gives me 6.99 inches, which we should of course round to 7 inches. That’s the distance you want to mark on the rods from each end (two marks per rod). With my 18-inch rods, I just lay them next to a ruler and mark at 7 and 11 inches.

Ok, now all is ready for construction. The basic step you’ll be doing over and over is placing two rods side-by-side and putting a rubber band around both of them and sliding it down to the first mark on the rods.

We’ll call this basic unit a “linked pair.” Take two linked pairs and put them side by side with their connections on the same side, and put a third rubber band around one rod from each pair at the ends opposite the existing connections. Slide the rubber band down so that you have a zig-zag chain of four rods.

Connect a fifth rod to one at either side, continuing the zig-zag.

Now connect the free end of this fifth rod to the free end of the first rod of the zig-zag, as shown on the right.

This is a configuration that you can now wiggle around until it forms a five-pointed star, as shown to the left. Important: looking at the two junctions on each rod, the rod must be underneath at the first junction and on top at the second junction as you proceed clockwise around the inner pentagon. Make sure that every one of the five rods follows this pattern before continuing, It’s this under-over, under-over pattern that constitutes the weaving in this construction, and that same under-over, under-over will continue throughout the entire construction.

Next, at each of the points of the star you’re forming, you can slip one rod under the other to continue the under-over junction pattern of the rods. These points of the star will eventually be vertices of the SSD.

Add a rubber band to secure each of the points, and congratulations, you’ve just made your first star out of 12.

To continue toward the SSD, take five rods and line them all up and put a rubber band around all five of them and slide it down just a short way, roughly the amount you decided to leave at the vertices above. You should then be able to splay the rods out so that the far ends are at the points of a pentagon, and there is a little clockwise swirl of the near ends of the rods (see the picture to the right).

Place each of the far ends just counterclockwise of one of the internal junctions of the star you made, in order around the star (as shown on the left).

Insert each of the far ends into the rubber band at the corresponding junction, making sure to keep it within the triangle of the star that it started in.

Now push all of the five new rods through those junctions until the second mark reaches the junction. That completes the first full vertex of your SSD.

If you look at your structure in progress from the side, you will see three out of five sides of another star. We want to complete that to a full star. So take a linked pair, and link one free end of the pair to each of the two sides of the star sticking out. Push those links in a little way and make sure the junctions all continue the under-over pattern of the first star. If all looks well, you can slide all of the connections until they reach the marks.

To complete the second star, slip the points of the star past each other as you did with the first star, and secure with rubber bands (there will already be rubber bands for this in place at two of the vertices).

Now rotate the structure 1/5 of a turn around the original star. You’ll see another star with only three sides in place. You want to complete this in a similar fashion. The only difference is that this time, you only make one entirely new link; at the other end of the linked pair you’re adding, you feed the rod through the existing rubber band at an internal junction. Make sure to continue the same under-over pattern throughout. The new rod should pass through a triangle in the second star that does not yet have a rod passing through it (it should not pass outside the second star or through the pentagon of the second star).

When that one’s done, add two more linked pairs to complete two more stars in exactly the same fashion.

The next linked pair is very similar to these; you just have to feed both of its free ends through existing junctions. When that’s done, you’ve finished six of the twelve stars in the SSD.

To continue from here, take a single rod (not a linked pair) and feed it through any of the partial vertices and the nearest internal junction that only has two rods at it, as shown.

Adding that rod will create a new partial star with three edges in place. Complete that one just the way you did the third star.

You’re getting close now. Complete two more stars like the third, and then one like the sixth.

That leaves just a single line of four junctions that does not have a rod going through them (shown down the middle of the image above). Carefully insert the final rod through all four junctions – it’s probably easiest to pass it through the two internal ones first, then tuck the two ends in at the vertices, to produce something like this. (This last edge is again down the middle of the image below.)

Now comes the exciting final part: cut away all of the rubber bands at the internal junctions. They were just there to stabilize things during construction; they’re not actually needed to hold the SSD together once it’s built, thanks to all of that weaving that we did. And voila!

In fact, if you used a high-friction material like bamboo strips or pipe cleaners, it’s often possible to cut away the rubber bands at the vertices as well. I don’t have a picture of this, as the metal rods are too slippery for the vertices to hold. If you want to try, just proceed carefully, and if any of the vertex junctions “pop,” tie it back up immediately and don’t cut any more (unless you want to quickly revert to a pile of separate rods).

If you like this construction and happen to be in Providence on 2019 Sep 28, come to the ICERM Open House at WaterFire Providence and help build a glowing version with 10-foot pipes for the rods!

12 Dec

Doing Math: The Math Behind Sand Piles

The mathematical purpose of this project is to explore the geometry of the piles that form when sand is poured on different 2D shapes. Specifically, we investigate the ridges and apexes that form on these piles. In the case of the polygonal shapes, we define a ridge as where two planes meet at a line, and an apex as where three or more planes meet at a point. We decided to focus on these aspects, and particularly their projections onto the 2D shape beneath them because they are easier to analyze in 2D, and because the actual angle the pile of sand slopes at depends on the sand itself, and not on the geometry of the shape it is on.

First, we define what convex/concave means. Consider any vertex of a polygon. Then look at the two vertices at the ends of the two edges that form the original vertex. If the line drawn between these two vertices lies at least partially inside of the polygon, then define the original vertex as a convex vertex. If the line lies entirely outside of the polygon, then define the original vertex as a concave vertex.

The main observation that we came to and hope that participants will gain intuition for is that, given a polygon, the ridges when projected onto the shape are the angle-bisectors of each convex vertex, and the apexes occur where these bisectors meet. Once they meet at an apex, the bisectors disappear and are replaced with a single new ridge that connects to another apex (this is particularly visible in the case of the rectangle). The logic behind this phenomenon is that a convex vertex produce ridges because sand resting on top of the shape needs to minimize the distance between its two forming edges, whereas with a concave vertex, its two forming edges create negative space where sand cannot rest.

Note that concave vertices do not create ridges or affect other ridges, which is exemplified by the the fact that the triangle and the concave hexagon have the same ridges. The logic behind this phenomenon is that a convex vertex produce ridges because sand resting on top of the shape needs to fall in the middle of two edges, whereas with a concave vertex, its two forming edges create negative space where sand cannot rest.

While the curved shapes appear to be more intimidating, they are in fact governed by the same principle: the “ridge” is the set of points equidistant to the two closet edges. In the cases of polygons, this set is the angle bisector.

In the case of the curved plates, this definition defines conic sections. For the circular plate with the circular hole in the center, the set of points exactly halfway between the two circles in a circle exactly halfway between the other two. For the off-center circle, this same process makes what looks like a circle in 2D, but is actually an ellipse in 3D. Lastly, the small hole on the rectangular plate simulates part of a parabola, since a parabola is defined to be the set of points equidistant to a point and and line (and here we approximate a point as a small circle, so sand can fall through).

19 Oct

The Sights and Sounds of Harvard Stadium

By Sara Bobok and Do Hyun Kim

Making Math Material

Fall 2018


  • Introduction


Figure 1. Harvard Stadium (Photo taken by Sara Bobok)

Pivotal to community building on Harvard’s campus is the Harvard Stadium, a space for sports, theater, and school spirit. Fascinating mathematics are necessary to allow for the Stadium to fulfill these functions. The photographs presented in this paper were taken of the Harvard Stadium on the week of October 1, 2018. Even within just a few photos, countless mathematical concepts can be observed. We will focus on the following: the acoustics of the amphitheater formed by the front of the stadium and the sightlines from differing locations. Through these topics, we were able to apply our interest in math to better understand the theatrical and structural aspects of Harvard Stadium. Understanding these topics can help us better understand how the Stadium serves its critical role on campus to ultimately enrich community building and student opportunities.


  • The Sounds –– Amphitheaters in History


For well over two thousand years, art and theater have played a pivotal role in civilization. The earliest performed play of which there are records, “The Persians,” debuted in the fifth century BC. Yet without modern amplification, early actors, directors, and theater enthusiasts alike faced a common problem: how would the audience hear the dialogue?

The amphitheater served as the answer to this dilemma. Due to the inherent acoustic properties of its geometry, spectators clearly heard all music and dialogue without modern sound equipment. How was this possible?

Figure 2. Amphitheater Section of Harvard Stadium with labels (Photo taken by Sara Bobok)

First, it is critical to consider the geometric design of the seating in an amphitheater. Declercq and Dekeyser study the Epidaurus theater in Greece in their critical paper to understand why the acoustics in this amphitheater are so clear. They utilize diffraction theory, or the theory that describes the ways in which light and sound waves change when met by (reflected or absorbed by) a surface, to study the interaction between sound and cavea (a technical word of the audience seating area) for amphitheater acoustics. Seen in the corresponding photographs, amphitheaters are strictly semi-circles. This corresponds well with sound leaving a source, which propagates in spherical wavefronts. Thus, sound waves generally reach each row of seating with similar characteristics (time, frequency, etc). Although the shape of the seats slightly obscures the perfect circular symmetry of the sound waves, this damping effect is not enough to be audible by a human ear. [4]

Figure 3. Bobok enjoying the acoustics of the Epidaurus Amphitheater studied by Declercq and Dekeyser (Photo taken by Rose Parisi)

Second, the material used to build amphitheaters greatly impacts the sound clarity. Amphitheaters are typically constructed using limestone. Limestone is a non-porous material and thus reflects, rather than absorbs, sound. However, even with reflected sound, what makes the speech clear and comprehensible? The answer lies in the uniform and geometrical design of the cavea. The key features here are the dimensions of the seats and the slope of the cavea itself. When present, the emitted sound is immediately reflected by the stage building and orchestra floor. The dimensions of the cavea allow for low levels of sound occlusion, which means nearly all speech comes across clearly. Furthermore, the correct frequencies for understandable speech are amplified. [4]

How does this study of acoustics apply to the Harvard Stadium? Geometrically, the Harvard Stadium fits the description of a Roman amphitheater: semi-circular in nature, thereby corresponding with the shape of the sound, and constructed with non-porous concrete material Although it is missing to features Declercq and Dekeyser identify as important––lack of a stage building and porous grass replacing what would otherwise be a limestone orquestra floor––the acoustics still work remarkably well. In the spring of 2018, the Harvard Classics Department and Office for the Arts collaborated with student director Mitchell Polonsky to perform an original translation of “Antigone.” Performing in most parts of the stadium, not just the front amphitheater required that actors have microphones. Yet, in one scene set close to the audience in the amphitheater, performed by actors Jacob Roberts and Aaron Slipper, Roberts and Slipper did not need microphones and were clear and audible by all of the five thousand spectators.

Thus, we find that the circular architecture of the cavea and the non-porous material are what allow the Harvard Stadium’s amphitheater section to provide clear acoustics for its viewers.


Figure 4a (Left): the musicians of Antigone performing with the use of microphones. Photo Credit: Richard Tong, 2018.

Figure 4b (Right): Roberts at the front of the stadium (which can be determined by his proximity to the “Crimson” lettering), acting without modern sound backing. Photo Credit: Richard Tong, 2018.


The Sights

In addition to thinking about how the seats were designed in relation to sound, it’s also interesting to see how the seats of the Harvard Stadium were designed to help the audience view the field. Like most other football stadiums, the seats are positioned such that each row is slightly elevated compared to the row in front. This is crucial for spectators to be able to view the football field, which is usually at the ground level. In other places, such as a church or a concert hall, seats in the back are not elevated as drastically, if at all. This is because the object the people are trying to view is elevated. An image demonstrates these key differences:

Figure 5. Stadium Seating vs. Standard Seating [6]

The Harvard Stadium was built to play American football. The most popular sport in America attracts millions of people to their TVs, peaking at over 100 million (slightly less than 33% of America’s population) during Super Bowl Sunday. The football program at Harvard is one of the oldest in the world (dating back to 1873; by comparison, the NFL was founded in 1920) and continues to host one of the most well-known rivalries in football history: “The Game,” against the Yale Bulldogs. “The Game” takes place every year, alternating between the Harvard and Yale stadiums.

Intuitively, it seems clear that the best seats would be somewhere near the 50-yard line. In fact, there is a blog that explains the “30/30 rule.”[7] In short, sitting anywhere between the 30-yard lines and close to the 30th row allows for the best view of the entire football field. Being able to view the entire field from the middle is much more advantageous than trying to watch the action from the ends of the field. This is because most of the action that occurs during football takes place near the middle of the field, and teams run from end-to-end. Here are some images that help support this notion:

Figure 6a. Side view of the Field

Figure 6b. Frontal View of the Field

As we can see, Figure 7a shows us the middle of the field very well, which makes it easy to see players run from one end to the other. On the other hand, in Figure 7b, it could be difficult to see what is going on when the ball is near the middle or on the other side of the field.

So the “between the 30-yard lines” rule makes sense, but how can we mathematically understand why the 30th row is the best? The aforementioned blog [7] goes no further to explain this notion, but we found that there are other concepts we can apply to explore this.

It turns out that we can assign a parameter known as the “C-value” to measure the quality of view.[8] Here is a picture provided by Professor Paul Shepherd at the University of Bath that depicts what the C-value is:

Figure 7. The C-Value

The higher the C-value (measured in mm), the easier it is for spectators to see the field. According to Professor Shepherd, a value of 120 mm is considered “optimal viewing standard.”[8] If the C-value gets too high, the spectator may have more of an “aerial view,” which is not considered to be an optimal viewing point. Professor Shepherd refers us to an equation he claims is used by engineers and architects that will be useful as we attempt to determine where the best seat in Harvard Stadium is.

Figure 8. The C-Value Equation

Figure 9. The Variables of the C-Value Equation

In this equation, N represents the difference in height between successive rows, R represents the vertical height from the field to a particular seat, D represents the horizontal distance from the field to a particular seat, T represents the depth of each row, and C represents the C-value.

Using fixed values for N, C, and T, we can try to derive an optimal “R” value that can help us determine the optimal row in Harvard Stadium. We’ll be using the following values, converted to meters (that we measured ourselves or used as standard values, such as for C):

N = 0.38 m

C = 120 mm = 0.12 m

T = 27.5 inches = 0.7 m

D = ?


Here, we run into an issue. It’s impossible to come up with a fixed value for D, because the value of D depends on which row we are considering, which is what we’re trying to solve. We know that the width of the field is 230 feet.[9] Thus, we can use trigonometry to calculate D’ (horizontal distance from seat to the edge of the field, where the bleachers end), and then simply add 115 feet, or about 35 meters. The hypotenuse can be found by using the Pythagorean Theorem, using “a” as the row number of the particular seat under consideration (in other words, the optimal row number we are looking for).

Figure 10. Updated diagram depicting D’, A, and hypotenuse

The equation for the hypotenuse is produced below:

This hypotenuse completes a triangle using D’ and R as sides. To find angle “A,” we realize this large triangle in Figure 11 is similar to the right triangle that can be created using N and T as the legs. From this, we can use the following equations:

We can use angle A to figure out the values of D’ and R in terms of “a.”

We recognize that R in this equation does not accurately depict R in Harvard Stadium. The first row actually begins at a certain height above the field. We estimated this height to be around 2 meters. Thus, we will use R’ in our calculations, where R’ = R + 2. We must also remember to use D (instead of D’), using the relationship D = D’ + 35.

Plugging in these values allows us to find “a,” the optimal seating row. I computed the value of “a” using WolframAlpha, which gave me an output of about 90. This means that the 86th row offers the ideal viewing point within the stadium. Unfortunately, Harvard Stadium does not have 90 rows, which brings us to a bit of a dilemma. Does this mean that there is no “optimal” C-value for any of the seats at Harvard Stadium? Or does this mean that a C-value of 0.12m is not applicable for Harvard Stadium? We discuss this problem later in this post.

We can try to gain some new insight by working backwards: we can use an estimated value for R (an R that corresponds to the 30th row, following the 30/30 rule) and see what C-value we obtain. Taking into account the height of the wall that separates the field from the bleachers, we can again estimate that to be about 2 meters before the first row begins. Thus, an approximate value for R, corresponding to the 30th row, is as follows:

R = [(0.38 m) * 30] + 2 = 13.4 m

We also need an approximate value for D. Again we can use the fact that we’re at the 30th row:

D = [(0.7 m) * 30] + 35 = 56 m

Let’s plug this into the equation to solve for C:

We obtained a C-value of 0.2m, which is well above the proposed optimal viewing value. In fact, doing the calculation for the 15th row gives us a C-value of 0.257, while the 1st row gives us a C-value of 0.327.

It may be useful to examine the possible assumptions of the C-value equation to better understand why we are obtaining such a high C-value for the 30th row seats at Harvard Stadium, and the C-value continues to increase as we go lower.  Although Professor Shepherd described the 120mm C-value as the “optimal viewing standard,” it might be more likely that this value is a minimum value for good viewing.[8] Intuitively, this makes sense because being able to have a larger C-value means that there the view is less obstruction by the person in front.

Knowing that a high C-value is good doesn’t explain why the 1st row at Harvard Stadium seems to be “the most optimal.” One can reasonable notice that people sitting in the first row may actually have a limited view of the field, since they don’t have a great view of the opposite side of the field. Perhaps there are some necessary conditions in order to use the C-value equation effectively. The mathematical calculations of the C-value support the notion that view obstruction is not a major concern at Harvard Stadium. It’s possible that the equation is more applicable for places where visual obstruction is more prevalent. Going through the concept and mathematical analysis of the C-value taught us not only where theoretically the “just optimal” seats at Harvard Stadium are (the non-existent 90th row), but it also seemed to imply that visual obstruction is not an issue at Harvard Stadium by showing that the 31 rows at Harvard seem to have a higher-than-optimal C-value.

Using concepts such as acoustics and C-values to analyze the design of the Harvard Stadium provided some insights into a place that many people visit every year. It helped us appreciate the engineering and architectural thoughts that went into the construction as well. Through mathematics, we can better understand the beauty that lies behind the structural and aesthetic components of the structures around us.


Sources and Acknowledgments

Special thanks to Richard Tong for providing the images in Figures 4a and 4b.

[1] Williams, Kim, & Ostwald, Michael J. (2015). Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s (2015 ed.). Cham: Springer International Publishing.

[2] Mourjopoulos, John. “The Origins of Building Acoustics for Theatre and Music Performances.”https://acoustics.org/the-origins-of-building-acoustics-for-theatre-and-music-performances-john-mourjopoulos/

[3] Miller, Melinda. “How Does Acoustical Absorption Work?” 22 Feb 2018. https://www.abdengineering.com/blog/how-does-acoustical-absorption-work/

[4] Declercq, N, and Dekeyser. “Acoustic Diffraction Effects at the Hellenistic Amphitheater of Epidaurus: Seat Rows Responsible for the Marvelous Acoustics.” The Journal of the Acoustical Society of America, vol. 121, no. 4, 2007, pp. 2011–2022.

[5] Farnetani, Andrea, et al. “On the Acoustics of Ancient Greek and Roman Theaters.” The Journal of the Acoustical Society of America, vol. 124, no. 3, 2008, pp. 1557–67.

[6] Stadium seating. https://en.wikipedia.org/wiki/Stadium_seating

[7] Hanson, Keith. “Where to Sit for a Football Game.” 4 Sep 2015. https://www.rateyourseats.com/blog/cheap_seats/where-to-sit-for-a-football-game

[8] Shepherd, Paul. “Sightlines.” 2012.


[9] Lottman, Michael.  “Nation’s Oldest Stadium Has Colorful Past.” 7 Nov 1959. https://www.thecrimson.com/article/1959/11/7/nations-oldest-stadium-has-colorful-past/.

03 Oct

Trees of Willow, True Love, and Tiling

Trees of Willow, True Love, and Tiling

Mai Nguyen

Math 168: Making Math Material

The “Willow Plate” design is a popular theme for china, here shown on its classic plate form.

I’ve been eating off these plates ever since I can remember. No matter how many beautiful or interesting pieces of dishware in our family’s house, I always gravitate towards the “Willow Plate” design.

My mom told me the story of the plate even before I started school: a high-ranking court lady and a low-ranking servant boy fell in love and wanted to get married, but could not due to the social hierarchy and control of her family. Eventually, they ran away from the altar where she was meant to marry someone else, escaping on boat to an abandoned island. Her father, furious, searched for them and eventually found out where they were. In desperation, the couple prayed to the spirits, who transformed them into a pair of love birds, and they were able to fly away together.

In the process of researching this lead, my inner Asian pride in the legacy of this beautiful blue-white design was shot down. The origins of the “Willow Plate” are in 18th century chinoiserie, plateware created and designed by Europeans inspired by China’s rich culture. So, not quite the authentic, Asian-pride narrative I’d hoped for. In any case, the plate leaves lots of imagination for mathematical exploration.

On the surface, the use of concentric circles can be outlined to create the plate. Although the Willow Plate design is very versatile and used on many patterns, the circular dish is most popular. I personally see the story with a distinct beginning-middle-end as a cycle of sorts. The use of concentric circles to outline the designs and create the eventual fluted shape and full plate emphasizes this.

Although the conception of the story’s meaning is simply a synthesis of European impressions of Chinese culture in the 18th century, the passing on of the story has transformed from a marketing gimmick to a point of connection at the dinner table for families and beyond.

In authentic Chinese pottery, geometric patterns and re-imaginations of nature are key points as well, but less as decoration and more to create a larger piece of nature.


Blue-and-white porcelain bowl with landscape, Izushi ware The Museum of Ceramic Art, Hyogo

In this plate, especially on the rim and outer sections, there is also symmetry by rotation. Around the rim of the plate, the patterns form small grids, bounded by the separate sections of the plate. Among them are many named tile patterns and others that I’ll describe myself, going from top to bottom:

  • Columns of a circle within a circle
  • Rows of a square within a square
  • Rows of a circle within square on the left
  • Ring of 6 trapezoids with curved sides and a hole cut out in the middle
  • Ring with inscribed superellipse
    • A superellipse is a closed curve contained within a triangle, resembling a curve-sided diamond
    • In this case, the resulting superellipse will look like a 4-pointed star with curved edges
    • The equation of a superellipse as \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=1, in the Cartesian form
  • 4-Petaled Flowers
    • petals inscribed with lines through vertices
    • The section between the flowers is, as above, a super ellipse with a factor of n (0,1), with a circle in the middle

Each of these patterns, when repeated, is used to fill up the plane. A tessellation refers to the filling up of a planar space with geometric shapes as a pattern, without gaps or overlap When the geometric pattern can be slid a finite distance and superimposed over itself, the tiling is periodic. Each of the cases above is periodic.


By contrast, an aperiodic tiling is one that cannot be slid a finite distance and be superimposed on itself. The Penrose tiling is a classic example. 

Sir Roger Penrose(1931-), an English mathematician whom the tiling is named after, became interested in tessellations mainly because of the artist M. C. Escher’s early works, exploring the possibilities of math and physics through art in finite space. A famous example of Escher’s work is the Devil’s Angel; in it, you can see similar concepts of a repeating pattern, a tile, filling up a space infinitely.

Fascinatingly, the golden ratio, φ (Greek letter “phi”), a special number = (1+√5)/2, ~1.618 and appearing many times often unexpectedly in geometry, art, etc., shows up here in rhombuses of the Penrose Tiling. One way to describe the golden ratio is that when you divide a quantity into 2 unequal parts, the ratio of the whole length over the longer length is equal to the ratio of the longer length over the shorter length.

In a regular pentagon, if you draw diagonals across the polygon, the ratio of the side length to the any line segment congruent to line segment EX will be φ (ex. line segment DE/ line segment EX= φ).

The obtuse triangle in the figure at left is congruent to half of the red rhombus in a Penrose tiling, and the acute triangle is congruent to half of the yellow rhombus. Thus, the construction utilizes the relationship of the sides of the tiles and φ.

Designs similar to the Penrose Tiling have been created in the past. For one, physicist Peter J. Lu of Harvard University claims that the quasicrystal, an ordered but not periodic structure, form of the Islamic “girih”, a geometric art form, has been found to very closely resemble the Penrose Tiling. While the design on the Darb-i  Imam shrine wall, built in 1453, is not completely perfect, it may be reminiscent of the Penrose Tiling. At the same time, the use of geometry to convey the perfection of God suits the golden ratio and its intrinsic beauty.

While the Willow Plate design does not utilize the Penrose Tiling, clearly there was geometric intent behind the use of tiling segments and symmetry. While the European makers created their own story and geometric imprints, the legacy of this type of dishware will always be associated with China. While history itself can be seen as linear and neatly chronological, my exploration of these plates was anything but. Similarly, any topic and idea in math can always be explored on a deeper level. When considering the existing foundation of knowledge, I continue to have the same sense of wonder at the never-ending paths for exploration in art and math alike.  I imagine that even the oldest civilizations felt the same way.


Sources and Acknowledgements:

“Cosmology, Escher and the Field of Screams.” In The Dark: A blog about the Universe, and all that surrounds it. Accessed October 3, 2018. https://telescoper.wordpress.com/2012/03/20/cosmology-escher-and-the-field-of-screams/.

Cut the Knot. “Golden Ratio in Regular Pentagon.” Cut the Knot. Accessed October 3, 2018. http://www.cut-the-knot.org/do_you_know/GoldenRatioInRegularPentagon.shtm. 

Google. “Blue and white porcelain bowl.” Google Arts and Culture. Accessed October 3, 2018. https://artsandculture.google.com/asset/blue-and-white-porcelain-bowl-with-landscape-izushi-ware/tAH1cc4-0EVE9Q. 

Johnston, Hamish. “Islamic ‘quasicrystals’ predate Penrose tiles.” PhysicsWorld. Accessed October 3, 2018. https://physicsworld.com/a/islamic-quasicrystals-predate-penrose-tiles/.

Meisner, Gary. “Penrose Tiling and Phi.” The Golden Number. Last modified May 13, 2012. Accessed October 3, 2018. https://www.goldennumber.net/penrose-tiling/. 

Wikipedia. “Penrose Tiling.” Wikipedia. Accessed October 3, 2018. https://en.wikipedia.org/wiki/Penrose_tiling.

———. “Tessellation.” Wikipedia. Accessed October 3, 2018. https://en.wikipedia.org/wiki/Tessellation.


03 Oct

Arches and Sangaku in Memorial Hall


Memorial Hall

Housing the largest undergraduate dining hall, a lecture hall, a student bar, classrooms, music rooms and more, Mem Hall is almost always bustling with student activity. Unlike other buildings on Harvard’s campus however, the architectural style of Annenberg Hall is unique, a fusion of New England brick with gothic arches, cathedral-like stained glass, and detail-rich ornamentation. Although a mathematical perspective can shed insight on many facets of the building, two areas of particular interest can be seen housed within the ceiling structure of the 9000-square-foot great hall. 

Although the exterior facade of Mem Hall is rich in detail, the wooden arches make the interior view even more exciting. To help better understand the vaulted ceiling from both a structural and aesthetic perspective, I will focus my observations on two distinct components: the central arches, and the circular geometric features which run around the perimeter of the space.

Interior Ceiling of Annenberg Hall

Unlike a number of arched structures on campus, such as the walkways through the dorms on Mass Ave, the arches in Annenberg are noticeably more pointed. On Harvard’s campus and beyond, arches come in all shapes and sizes. Engineers in the 21st century consider a number of factors when choosing the style of arch best suited for a structure. Every arch design has its own unique structural properties, which makes each design optimal in different settings. Beyond just structural identifiers however, each arch is also unique in its cultural identity and assembly technique.  Although modern technologies make it possible to assemble a huge range of arches, when Mem Hall was constructed almost 150 years ago, hand-drafting techniques and less advanced building technologies presented some constraints. Rounded arches, which were popular in Romanesque architecture, distributed weight through barrel vaults connected directly downwards onto external walls (think old European cathedrals with really thick walls and tiny openings for windows). The three-point arch, heavily popularized in gothic architecture–  which can be seen running though the central cavity of Annenberg– has, in comparison, a far greater load capacity with an ability to distribute weight outwards, rather than only downwards.

Different styles of arches.

Three Point Arches in Annenberg.

If you’re interested in learning more about different types of arches, and when different styles are most appropriate, any structural engineering book like this would suffice. You can also check out some cool online resources to garner a better sense of how the form of arches relates to materiality as well.


Circular Detail in Annenberg Hall

Although the arches are perhaps the most noticeable feature of the roof system, it’s the collection of a tremendous amount of detailing and craftsmanship that makes Mem Hall so breathtaking. Hiding on the outer perimeter of the arches are a beautiful series of wooden cutouts, less reminiscent of gothic architecture than they are of  Sangaku, an ancient Japanese art form involving puzzles that find relationships between circular geometries. Not only do Sangaku works often showcase beautiful designs and geometries, but they can also highlight unique properties of circles and other two dimensional shapes.

Example of Sangaku

Before looking too in depth at the geometries within Annenberg, it might be useful to look at an example Sangaku problem. Take, for example, the challenge to arrange  three circles such that all three geometries share a common tangent line.  Below is one solution to this challenge, but are there others? Note in the solution below other interesting elements as well, such as the fact that the radii of the two smaller circles, when multiplied together, equal the radius of the larger circle. Sangaku puzzles often introduce interesting relationships such as this one, and in turn, lend themselves to explorations about a number of geometric relationships. From this prompt alone, one might ask if the relationship of the radii found in this solution will always hold if there are in fact other solutions.

Sangaku Example Problem

Although Sangaku problems are typically driven from a proposed conjecture, we can also work in the other direction (by first looking at a geometric design) in an attempt to better understand the wooden cutouts in Annenberg.

Overlying Circular Geometries in Annenberg’s Wooden Cutout

Simply by overlaying circular geometries over the wooden cutout, we already are able to observe some interesting relationships. Most immediately apparent is that when the four white circles are inscribed within the larger red circle, the radius of the circle that can be drawn in the middle of the entire geometry is equal in radius  to the circle that can be drawn between the outer red circle and two of the larger white circles (in reference to the picture, look at the two yellow circles).

In this picture, the 4 inscribed white circles all have equal radii, but after seeing this drawn out, I became curious about other ways of inscribing 4 sub-circles into a larger circle, and what such arrangements possibly dictate about radii length. Here, our 4 sub circles have combined area 4*pi(4)^2, or about 200 square units relative to the area of the red circle, pi(9)^2, or about 254 square units. We can easily see here that the tangentially inscribed white circles are therefor about 4/5ths of the area of the red circle. Using this fact alone, I was curious if I could make a conjecture about  upper and lower limits of the area for 4 inscribed circles within a larger circle. To focus this problem a bit more, let’s keep the necessity for the circles to be tangent at the 12-3-6-9 o’clock positions.   To visualize what I’m describing here, here are just a couple possible interpretations of this problem:

Alternative ways of inscribing circles

Although my inclination was that the smallest ratio of the area of the inscribed circles to the exterior circles would have come with all 4 inscribed circles having the same radius, the example on the left is approximately 3/5ths, and on the right is nearly 1:1 (in the limit case where the large circle grows even larger and the three smaller ones approach radius 0). It turns out that the ratio is smallest when one of the inscribed circles has a radius approximately  3/4ths of the radius of the exterior circle, one has a radius 1/4th the length, and the other two circles have equal radii. In the limiting cases, this puts the lower limit of the ratio at approximately 1/2, and the upper limit at 1/1.

Using a mathematical perspective to analyze the design of Annenberg hall not only gives us a chance to understand the building in a new light, but also gives us a chance to imagine the design process that went into such a structure. In a sense, mathematics equally played a role in the structural and aesthetic construction of the building, and just by looking up and around, we have a chance to better understand the arches and ornamentation that contribute to making the building the landmark it is today.


Sanders Theatre Harvard University, Photo by Chensiyuan, courtesy of Wikipedia.

Jacob Bindman

A Sangaku dedicated to Konnoh Hachimangu (Shibuya, Tokyo) in 1859, courtesy of Wikipedia.

Sangaku puzzle in which three circles touch each other and share a tangent line, Tibbets74, courtesy of Wikipedia.

Model built using Adobe Illunstrator.

03 Oct

Illuminating Angles: Preserving Artwork Through Math

Lacey Hines and Katja Diaz-Granados

The Quincy Street entrance to the Harvard Art Museum.  Image taken on October 3rd, 2018.

In 2010, Harvard University broke ground on the three year long renovation project to merge its three art museums – The Fogg Museum, the Busch-Reisinger Museum, and the Arthur M. Sackler Museum – under one roof. Architects not only faced the difficult task of building a structure large enough to fit such extensive collections, but they were further constrained by curators’ unique desire to exhibit paper-based drawings and canvas paintings side by side in the same gallery.  For those unfamiliar with art conservation, it may seem like common sense to show all of an artist’s works together side by side, no matter the medium. And while art is art, the varying chemical structures of art media and materials mean that different pieces of art react in different ways to sunlight, temperature, and moisture. Because of this, paintings, paper-based drawings, and sculptures are historically displayed in separate galleries with individualized environmental settings.  To design a space that would allow paintings and paper-based drawings to safely coexist, Harvard University hired Italian architect Renzo Piano.  His solution was the piéce de résistance of the museum: the glass lantern roof.

External View of Museum Building (Ryan 2014)

The roof is a glass, pyramidal structure composed of four sides of temperature and light controlled glass panelling. Members of the museum staff are able to adjust the amount of light coming in through the roof using a small remote device. While all four sides of the roof are visible from the outside, on the inside, the main focal point is the slanted expanse of glass over the Calderwood Courtyard within the entrance of the museum. This courtyard is where visitors purchase their tickets or can sit and enjoy refreshments from the museum cafe. From the seating within the center of the courtyard, one can gaze up and see the upper level of the museum’s galleries bathed in light. 

Glass roof as seen from the Calderwood Courtyard. Image taken on October 3rd, 2018.

The skylight transmits light across the galleries, acting as what Piano calls “a light machine”. The museum is centered around the Calderwood Courtyard with galleries and arcades lining the four sides of the courtyard on each level. Upon entering the museum, it’s impossible to miss the rays of sunshine streaming in, and yet a tour guide will quickly tell you that the rays don’t come into contact with the fragile artwork a floor below the sculpture gallery. This begs the question, is there something special about the angle of the roof that allows the sun shining through to reach only the bottom and top floors? And assuming the angle of the roof was chosen specifically for the light-directing feature, is there some other angle that could accomplish the same task?

To approach the question of what angles allow light to hit the interior courtyard and sculpture floor, yet spare those arcades with painting and paper galleries, imagine the profile of the interior courtyard as an irregular hexagon.  The six sides consist of the floor, the walls of the courtyard and the three hypothetical panels of glass for the roof.  The angles for the floor and the walls must be constrained (since we are assuming the walls meet the floor at right angles).  However, there is potential flexibility when it comes to the angles forming the roof itself.

Light is transmitted optimally when a glass pane is oriented perpendicular to the light.  According to Linsdey Schiller, a researcher interested in greenhouse design, a  90 degree relationship between the light and the glass allows a full 90 percent of light to be transmitted through the glass (with the rest absorbed as heat).  Any angle higher than 90 degrees results in some light being reflected off the glass or additionally lost from light absorption.

To know what angle of incidence corresponds to the light rays in this way – hitting the glass full on – you need to know the angle of the sun for your particular location, a parameter that varies with both latitude and season.  A conservative estimate would mean knowing the angle of the sun in the winter, since the winter sun tends to be lower in the sky, and so any calculation involving this lower angle will produce a slightly steeper pitched roof.

Architects have often relied on the simple calculation of adding 15 to 20 degrees to the latitude of their build location to get a quick estimate of the angle that would be perpendicular to the winter sun.  The question then is, is this the only angle?  Schiller and her group studied the percent light transmitted with changing angles of incidence, and found that there is in fact a range of ‘good’ angles.  From that initial ideal angle, angles up to 50 degrees lower won’t have a significant effect on the overall light transmission, typically changing light transmission by a few percent.

Schematic of optimal angle of incidence. (Schiller 2015).

The curious plateau of light that can be transmitted across a range of low angles and the steep decline in transmitting ability at steeper angles, suggests that it’s the detrimental effect of light reflection and loss due to light absorption that is governing the relationship between angle and light penetration.

Since Cambridge has a latitude of approximately 52 degrees, the range of angles for the two sloped roof panes of the museum should be between 22 degrees (52+20-50) and 77 degrees (52+20).  The actual angle of one of the museum roof panels was measured using imaging software to be 24 degrees, which falls nicely within this range.

Sketch of the art museum. Arrows indicate potential light transmission as the sun migrates across the sky.

To understand how the different sloped panels of the roof each individually contribute to the lighting we can think of the cross section of the building as having certain areas where we want safe from light.  Since paper and painting artwork are stored on the first, second and third floors, they determine the areas where light should be excluded.  In most buildings this would not be an issue, given that light from the roof would only illuminate the top floor.  However, the open courtyard means light from the roof has the potential to reach galleries on lower floors.  Because the height below which light should be excluded is the same on either side of the building, an aspect of symmetry is needed to ensure light hits either wing of the sculpture gallery optimally, without encroaching on the floors below.


Exterior of the Harvard Art Museums highlighting the angled glass roof. Image taken on October 3rd, 2018.



03 Oct

The Behavior of Concrete Stripes and Lattices

Harvard’s new Smith Campus Center opened less than a month ago, and ever since then I’ve done most of my studying there. While the main Harvard Commons area is all brand-new wood and glass, the building’s main corridor is dominated by large, exposed concrete walls. While exiting the building one day, I noticed that these walls have a particular and peculiar design to them which consists of parallel lines and rows of holes spaced two feet apart. Below are a few pictures of the concrete that I took last week. These lines are either vertical, horizontal, or measured to be at 45 degrees.

My first question was why the walls has this pattern as opposed to just being flat concrete. After doing some research online, I found out that these “holes” are a by-product of the process of pouring the concrete. When a wall of concrete is poured, a mold must first be set up. If you simply pour the concrete in a box, however, due to the weight of the concrete and its inclination as a liquid to spread out, the concrete can break the mold and ruin the wall. In order to keep the mold together throughout the process, metal rods called “snap ties” are put through the mold. As seen in the picture below, they are supported in the middle of the mold by reinforcing bars, which are tied together in a square grid. This explains why the holes are in rows and columns instead of, say, a triangular grid. In order to be the most efficient, they are evenly spaced, except possibly near the end of the wall if there isn’t enough room. This can be seen in the first picture above. When the concrete is poured, the rods hold the walls of the mold together and resist the outward pressure of the wet concrete. Once the concrete has dried, a cone mechanism tying the rod to the mold can be snapped off, which frees the rod and leaves the hole (and explains why they are called snap ties).

credit: https://www.youtube.com/watch?v=1xRN4T3XjdQ

The stripes, on the other hand, are likely intentional. According to this architecture article, the default is to use big sheets of plywood for the concrete mold, as shown above. Since the sheets are flat, they produce flat, concrete walls. However, a more stylistic approach is to make something called “board-formed concrete.” Instead of using just plywood, A plywood mold is made and then the inside of it is lined with wooden boards. Then, when the concrete dries, the imprint of the boards (and the lines between them) remain, giving the wall a cool pattern, which is the case in the Smith Center. The stripes I observed were likely either horizontal, vertical, or at 45 degrees because these cases make setting up the lining easiest and have the most symmetry. Since this extra lining of boards adds time to the construction process, it must have been a aesthetic rather than a practical decision.

the inner wooden board lining. Credit: https://www.lifeofanarchitect.com/board-formed-concrete/

Here we have two separate phenomenon going on: one practical and necessary for making the wall, and one aesthetic and optional. Yet together they form two objects very dear to mathematicians: points and lines.  The points themselves, since they are evenly spaced in rows, form an integer lattice. An integer lattice in 2d is simply the set of all points (x,y) in the Cartesian plane where x,y are integers. The lines on the wall naturally translate into lines on the cartesian plane, and the question that first arises is if these lines intersect lattice points. This problem, however, ends up not being very fruitful. If a line through the origin has irrational slope, it will not intersect any other lattice points. If it has rational slope p/q, it will intersect any lattice point that is a multiple of (q,p). (If you are interested in learning more about this question, this is a good source).

an integer lattice (source)

Initially, the walls did not so much cause me to wonder, but rather annoyed me. Since the holes  and the lines are products of unrelated processes, the lines did not align well with the lattice, giving the walls an aggravatingly irregular pattern. And it was less about the lines intersecting or not intersecting the holes, but rather how the area between the lines intersected the holes. That is, it felt natural to think of the area between two parallel lines as a “stripe” and see how many holes a stripe had in it. As you can see in the photos, some had none, some had one, and some had two or more.

The mathematical problem I came up with relating to this was: which stripes in an infinite integer lattice have to contain a lattice point? Which don’t?

It’s trivial that generally horizontal and vertical stripes don’t have to contain a lattice point (just place the line between two rows or columns of lattice points!). Note that to simplify calculations I will sometimes refer to a stripe’s “vertical width” instead of its width in the traditional sense. The vertical width the vertical distance between the two parallel lines determining it. Using some trigonometry, you can convert this vertical width to regular width as desired. Note that vertical width is undefined for vertical stripes (which will be okay since we will not be dealing with this case anyway) and that vertical width can often be misleadingly larger than regular width, particularly when a stripe’s angle approaches π/2.

A first significant observation I was able to come up with was that all stripes with irrational slope contain a lattice point. This follows from the fact that integer multiples of an irrational number modulo 1 are dense (by modulo 1, we mean we take only the decimal part of a number, and not the integer part, leaving the result between 0 and 1). A good proof of that fact is here. That is, if we take an irrational r and take the multiples of r: r, 2r, 3r, etc., there will always be one of these multiples between any two given numbers between 0 and 1. Because of this, if we set our stipe to be centered around a line y=rx+b with vertical width ε, at integer x-coordinates, the corresponding y-coordinates are dense in [b, 1+b] mod 1, which is the same as being dense in [0,1] mod 1. That is, there will always be an y-coordinate on the line within ε/2 of an integer, and thus the stripe will contain that lattice point.

Now that we have covered the irrational slope case, we can look at rational slopes. As for the second result, given the line y=(p/q)x+b such that p/q is rational in lowest terms and if b is not an integer multiple of (1/q), there exists a stripe centered around the line that does not intersect any lattice points. For the proof of this, consider the set of y-coordinates of the line when x-coordinates are integers: p/q+b, 2p/q +b, 3p/q+b, etc. Note that we only care about these values mod 1, since this determines their vertical distance from a lattice point. In this list, once we get to q(p/q)+b= b (mod 1), the list starts over again. Thus, this list is finite mod 1. Moreover, since b is not an integer multiple of (1/s), none of these numbers are 0 mod 1. Thus, since each of these numbers is a finite, nonzero, vertical distance from a lattice point, there is a minimum vertical distance, and thus a stripe with vertical width shorter than double that minimum distance will not intersect any lattice points. We can convert the vertical width of that stripe to regular width.

If the y-intercept is rational, we can actually easily find this regular width (we use regular width in this case since we are going to find the closest lattice point to a line, and regular distance from a point to a line corresponds to the regular width that stripe can be.) The proof is influenced by this post and this post, both of  which give a similar method for finding the closest point to a line in a lattice. If the slope and y-intercept can be written as rational numbers, then the central line has an equation of the form ax+by+c=0, where a,b,c are integers. However, we are later going to want a,b to be coprime, so we are going to divide out any common factors, leaving us with a line with equation ax+by+(p/q)=0, where a,b,p,q are integers, a and b are coprime, and p/q is in lowest terms. Using the distance formula from line to point the distance from this line to a lattice point (s,t) is |as+bt+p/q|/√(a^2+b^2). Note that there’s a theorem in number theory called Bezout’s Theorem from which it follows that because a,b are coprime, there exists a linear combination of a,b that is equal to one (aka there exist integers n,m such that 1 = an+bm.) If we multiply both sides of this equation by any integer, we can get an integer combination of a,b that is equal to any integer (i.e. d=d(an+bm) =a(dn)+b(dm)). In particular, we want to find the lattice point (s,t) such that as+bt is as close as possible to -p/q (making the numerator of the distance formula above as small as possible). But for that we can simply make as+bt equal to the integer closest to -p/q, which then makes the numerator the distance from p/q to this closest integer, which is min(p/q mod 1, -p/q mod 1), where again mod 1 gives a decimal part between 0 and 1. Thus, the smallest distance between the line and a lattice point is min(p/q mod 1, -p/q mod 1)/√(a^2+b^2) (which should be nonzero according to the proof above), and thus any stripe with regular width less than twice this distance will not intersect any lattice points.

While I did not come up with or find a good way to get this distance when the y-intercept is irrational, the above method could be used to estimate or get some kind of bound on this (by estimating the y-intercept with a rational number). The Diophantine approximation can be used to find the best approximation of an irrational by a rational and also bounds its error.

Since the stripes I encountered in the Smith Center were either horizontal, vertical, or had a rational slope of 1 or -1, if we extended the lattice and stripes of these walls indefinitely it is possible that they contain stripes that do not intersect a lattice point. But as we saw above, just a small shift to an irrational slope guarantees all stripes contain a lattice point. I wouldn’t have guessed that something as mundane as concrete walls could lead me to such an interesting result.

02 Oct

The Curvature of Hammocks

Hammocks often symbolize ease, relaxation and simplicity. Yet the mathematics of these structures contrasts with their seemingly effortless nature.

The image above shows a hammock in Quincy courtyard, captured on a Monday afternoon.

The particular Quincy courtyard hammock shown above consists of a rigid curved stand, and knotted netting attached to horizontal wooden bars. The wooden bars are in turn connected via more netting to hooks that hang on some fixed endpoints that are attached to the curved stand (or broader structure). Overall, hammocks can vary with regard to design (see the red hammock with an angular stand in the background), but the general structure of a membrane hanging between two rigid lines or points remains consistent.

Lying in this hammock, I started to wonder about the mathematical dynamics of the membrane supporting me. I became curious about two specific aspects of the hammock. One aspect is the natural curve of the hammock when it hangs in a resting static state without additional weight. The second aspect I am interested in is how that curve changes when the force of a body is added.

The underlying math of a hammock’s curve – both with and without added weight – are quite valuable to comprehend because they are reflected in other everyday structures such as arches and bridges. In this way, understanding the math of a smaller, everyday item such as a hammock can transfer to understanding the math of larger structures that carry more weight (literally and metaphorically) in society.

An initial glance at a hammock’s outline – viewing from a side perspective – reveals that there is an inherent curvature to the shape of an unloaded hammock at rest. To try to describe that shape geometrically, one could start by saying it resembles a crescent moon. When attempting to fit an equation to the curve formed by the hammock, however, it is important to take into account the nature of the membrane and the hammock’s construction. To begin, we can treat the curve as if it were two dimensional (later on we will ponder how the actual three dimensionality of the hammock could change the equation of the curve). From an initial glance, one might guess that the curve of a hammock is a parabola, with the equation y = cx2(where c is some constant). However, as it turns out, an undisturbed hammock actually takes on the form of a catenary curve, precisely because there is no added weight.

A roughly side-on-view of the hammock, showing the 2D curve of the structure

A catenary curve is the curve of a hanging chain hanging under its own weight when supported at its ends. In contrast, a chain forms a parabolic curve when under a horizontally uniform load. Algebraically, a catenary curve is expressed by the equation y = acosh(x/a), where cosh is the hyperbolic cosine function. The diagram below shows the difference between a catenary and parabolic curve. While subtle, the parabola rises more quickly near the vertex, and more slowly farther away.

 Visualization of a parabola and catenary – https://www.intmath.com/blog/mathematics/is-the-gateway-arch-a-parabola-4306

To better understand how this difference in curves occurs mathematically, we will explore how the equation for the catenary curve is derived:

The weight (W) is proportional to length of the chain between O and P, so we need to determine the arc length of the chain.

Let: y = height of chain, µ = linear weight density, s = length chain between O and P

In contrast, for a parabolic curve (such as a suspended bridge with a roadway underneath), the weight of the roadway is evenly distributed and so the weight density is linear. Since the road is flat underneath, the length of the road between two points is the distance between the x coordinates. This linear distance reflects the arc length and in turn integrates to a quadratic function. See below the derivation of a parabola:

So far, we have relied on a two-dimensional approximation to model the hammock. This raises the question does the edge of an actual three-dimensional hammock actually form a catenary curve? Well, the catenary curve already exists in everyday life in other objects that span three dimensions. However, the paths in those objects are transversely rigid, and so the objects are constrained to be a projection of a two-dimensional curve. For example, there are simple suspension bridges (also known as catenary bridges) where the path follows the cable. There are also catenary arches, which are used to guide the placement of building materials. A hammock is uniquely different in that the center of a hammock hangs lower than the sides. As a result, we want to explore if that difference in hanging distorts the catenary along the sides or not.

We begin by capturing a picture of the hammock from a side-on view without any distortions based on the angle of the view. In this picture, we make sure that the two “endpoints” (locations where the netting meets the chain attached to the frame ) appear to be the same height. The lowest point is then horizontally midway between those two ends. We can calculate a catenary curve that goes through three points – the two endpoints at the same height and a minimum point halfway between those two endpoints. Unfortunately the original white hammock was not available at the time I took the picture, so I used the red hammock (which exhibits the same properties). The resulting catenary plot looks like this:

It is clear that the catenary curve is a great fit for the free hanging hammock. We can then move the curve to a different y intercept and see that it follows the edge of the hammock.

We can clearly visualize how the catenary curve applies to a hammock at rest in multiple locations (such as along the edge). This discovery leads to the conclusion to our original question that the edge of a three dimensional hammock does indeed form a catenary curve.

But hammocks are meant for – and much more enjoyable when – lying in them. How will a person’s added weight change the curve of a hammock? A hammock is meant to carry additional weight, and its membrane’s shape changes when weight is added. The material of the membrane is often flexible to accommodate the extra weight, which in turn causes the catenary curve to be elastic. However, with enough added weight, the initial catenary curve can lose it’s shape and can rearrange itself in space. One possible final formation is a parabolic curve.

Elastic Catenary as it stretches: blue is the catenary and green is the parabola -https://www.mathcurve.com/courbes2d.gb/chainette/chainetteelastique.shtml

In this way, we can guess a hammock holding a person will form a parabola. Unfortunately, a person does not represent a horizontally uniform load, rather a person’s weight can be concentrated at his/her center of mass (slightly below the belly button). As a result, when someone rests in a hammock, the weight will be added to the middle region of the hammock netting (aligned with the body’s center of mass), causing the hammock to dip lower near the center. Unfortunately, there was not enough time to take measurements and run actual calculations to determine the shape of the curve of a hammock with a body resting in it. The calculations become even more complicated because a specific body’s weight and resting spot on the hammock can further affect the shape. There is much to be discovered, and understanding three-dimensional hanging membranes (and how they change shape with additional weight) is a subject of ongoing mathematical research.

To sum up, some may consider it counterproductive to think extensively about the mathematics behind the hammock, especially considering that the purpose of a hammock is to relax and ease the mind. However we should remember that taking a moment to contemplate the mathematics of a hammock can provide a new level of appreciation for the object that provides much needed support.


Resources and Acknowledgements:

Ferreol, Robert. Elastic Catenary. 2017. https://www.mathcurve.com/courbes2d.gb/chainette/chainetteelastique.shtml

Carlson, Stephan C. Catenary Mathematics. https://www.britannica.com/science/catenary

Kunkel, Paul. Hanging with Galileo. 2016. http://whistleralley.com/hanging/hanging.htm


Gambino, Stephanie. Center of Mass of a Human. The Physics Factbook. 2006. https://hypertextbook.com/facts/2006/centerofmass.shtml

30 Sep

Perfect Alignment in Reflections

Overview of the exhibit WeltenLinie (One in a Time) – by Alicja Kwade

WeltenLinie (One in a Time) – by Alicja Kwade

WeltenLinie (One in a Time) – by Alicja Kwade

IMG_5513, IMG_5526 

Videos of the exhibit WeltenLinie (One in a Time) – by Alicja Kwade

These images were taken during the summer of 2017 when I attended the Biennale in Venice on an exhibition in the main pavilion section, “Viva Arte Viva,”  of the Biennale. This exhibit – WeltenLinie (One in a Time) – by Alicja Kwade uses powder-coated steel, mirror, stone, bronze, aluminum, wood, and petrified wood. The objects are placed on either side of mirrors so that as the viewer walks past, the actual object is replaced by the reflection of another, perfectly lining up as the viewer moves past to create the illusion of a whole object, except with different materials making up each the reflected and actual objects.

This exhibit relies on symmetry and angles of reflection to create the perfect alignment that leads to one object being replaced by the reflection of another as the viewer walks past. Therefore, as the viewer moves, the angle of reflection and the amount of each object that the viewer is seeing changes. The diagram below shows an example of an angle where the viewer sees approximately half of the “real” object and approximately half of the reflected object. The arrows indicate the viewer’s motion and the resulting change in these angle relationships and what the viewer then sees after this movement. As the viewer moves to the right, he or she sees more of the original object and less of the reflected object. The lighter lines in this diagram generally reflect the result of that movement. Note that this diagram shows a view from above to highlight the angle lines. However, the objects will be viewed from a side view, not from above, in order to be actually reflected. Therefore, there is some error in how these objects are being represented in this diagram.

The mathematics behind this exhibit include issues of symmetry and angles of incidence and reflection with the mirror. The angle of incidence is the angle between the perpendicular line intersecting the mirror and the line between the object being reflected and the mirror. The angle of reflection is the angle between the perpendicular line intersecting the mirror and the line between the viewer and the mirror. These two angles are labeled in the above diagram. These angles must be equal because of the physical properties of a mirror and how light is reflected off of mirrored surfaces. Therefore, as the viewer moves to the left, these angles increase (also shown in the above diagram with the lighter lines). 

Below is a specific example of how the angles of reflection and incidence line up in this exhibit. The viewer is located at point C and the mirror is line f. Poly1 is the object being seen directly by the viewer. Poly2 is the object being reflected then seen by the viewer. And Poly3 indicates the combination of Poly1 and Poly2 which the viewer is seeing. The viewer is only seeing half of Poly1 – the DGF half – which corresponds to the ONM half of Poly3. The other half of the final object the viewer is seeing is the HKJ half of Poly2, which is being reflected in the mirror. The lines from Poly2 to the mirror indicate the reflection and the angle created between these lines and the perpendicular to the mirror are the angles of incidence. In this case, the angles are 45º. The resulting lines from the mirror (line f) to Poly3 indicate the resulting reflection. These lines along with the perpendicular to the mirror form the angles of reflection, which are also 45º.

The key to ensuring this alignment and perfect reflection is that the two objects must be perfectly lined up on either side of the mirror. Additionally, the one in front of the mirror (whose reflection will be seen by the viewer) must be a mirror image of the opposing object so that when reflected, the two objects align. This is easier to see in the original diagram above because the objects are not squares (as they are in the second diagram).

The mathematics described above enables the playfulness of this exhibition because the alignment of the objects and the mirror nature of them must be perfect in order to trick the viewer into seeing a single object created by a reflection and an actual object. In this sense, the math enriches the exhibit because, while it is not overt to the viewer (the angles aren’t labeled, the physics involved isn’t described in the exhibit, etc.), it is by necessity present in the exhibit due to the nature of mirrors, light reflection, and the creation of pairs of objects that are mirror images of each other.

This exhibit made me wonder what would be needed to recreate this effect. Additionally, I wanted to try having lettering on the objects. I did a simple recreation the setup using a makeup mirror and two identical bottles of nail polish containing different colors of nail polish to distinguish them from each other in the final image. The image below shows this setup, with lines showing the reflection of the purple bottle and the direct viewing of the red bottle by a viewer to the right. The dotted lines represent the mirror and the perpendicular lines to the mirror line, showing the equality between the angle of incidence and the angle of reflection. 

The image below shows this reflection from the viewer’s perspective. As you can see, only the end of the purple bottle is visible, with the majority of the red bottle being seen directly by the viewer.

As the location of the viewer changes, the ratio of the amount of purple versus red bottle changes. As seen below, the viewer is now further to the left and more of the purple bottle is visible through the reflection.

The image below tries to demonstrate this change in perspective. The angle is not quite the same as the above image but there is a noticeable difference between the first instance of the aerial view and this one. The angle of reflection and incidence has changed with the changed location of the viewer (who has moved to the left).

Finally, lettering on the object being reflected would have to be printed in the mirror image, so that when it is reflected, it is legible and lines up with the object being seen directly. In experiments with the nail polish bottles, the lettering did not line up between the reflected object and the direct object. The mirror flipped the letters. Therefore, in order for this to be perfect, the lettering on the bottle being seen in the mirror reflection would have to be printed as the mirror image of the original lettering. Only then would the bottles and lettering line up between the reflected and direct objects. Ultimately, it was relatively easy to recreate a simple setup of the original exhibit by Alicja Kwade. The beauty of the exhibit comes from the forms chosen and the clean setup of the objects and mirrors. When first seeing the exhibit, it is not obvious to the viewer that the reflection trick will occur as he or she walks around the exhibit. This is discovered as the viewer explores, drawing the viewer’s attention to the objects, reflections, and, indirectly, to the mathematics behind this alignment. 

For more information on this exhibit and the Biennale, visit:



30 Sep

The Math of Your Breakfast Bagel: Toruses and Mobius Strips

The inspiration for this blog post is a plain breakfast bagel, as pictured below in this photo taken in Winthrop Dining Hall on a Sunday morning. The bagel closely resembles a shape very important in the topics of geometry and topology in mathematics–that is, the torus.

The Bagel and the Torus
The torus is a surface which you can picture as follows:

Tori can live in any n-dimensional space for n>2, but because humans view bagels in 3-D, this discussion will center around the torus constructed in 3-dimensional space.

One way to visualize the torus is as follows:

  • Consider a toothpick held vertically against a tabletop. This will be our axis of rotation.
  • For a circle drawn on the same plane as the line passing through the toothpick, the surface created by spinning the toothpick 360 degrees forms the entire torus.

However, bagels are not hollow. Therefore, the rest of this discussion will focus on the solid torus, whose surface is an ordinary torus like the one visualized in the steps above, but also includes all of the points trapped within the torus.

Even acknowledging that real bagels are imperfect and therefore only closely approximates the solid torus, we see that this seemingly mundane and rather plain breakfast item becomes a foundational example in the field of topology!

Cutting The Bagel

If you enjoy your morning bagel with cream cheese, then some interesting mathematics might come into play: How should I slice my bagel so that I can have more space to spread cream cheese?

Typically, the bagel is sliced horizontally to create two halves on which to spread cream cheese. Let us say we will limit the slicing to one connected motion of a knife.

What if we slice using a cut shaped like a sine wave all around the bagel? Surely that would give more surface area of exposed white bread for cream cheese.

Expanding this to an extreme case, what if we keep increasing the frequency of this sine wave so that we have a near-infinite number of ridges on which to spread my cream cheese? Continuing this logic, in theory there is no limit to the amount of surface area we can generate with the cut. Therefore this question becomes difficult to answer. Perhaps the best way to maximize the surface area of exposed bagel is to just cut as many ridges as physically possible with my knife along the side of my bagel, where the limit of how thin we can slice is only physical reality.

An Interesting Cut

In 2012, Professor George W. Hart published an article about an activity he introduced to his students at Stony Brook University in which they were challenged with slicing a bagel into congruent, interlocking halves in one cut.

If you want to give this puzzle a shot, then put a hold on reading this blog post and find yourself a bagel and a knife.




The solution to this puzzling task is to slice the bagel along a two-twist Mobius Strip. (https://www.georgehart.com/bagel/bagel.html)

*One great way to visualize a two-twist Mobius Strip is to take a strip of paper, turn one end around twice while holding the other still, and gluing the two ends together.

I bring this up because Professor Hart’s challenge to his students is a fun and engaging activity that draws the attention of people who may never have even heard of tori and mobius strips. The idea of a single cut that splits one ring into two interlocking ones is exciting, and for that reason I will use this specific cut as an example in the exploration of surface area. Does this cut increase the amount of exposed bagel on which I can spread cream cheese? While this is surely not the most time-efficient method of cutting a bagel, we can use mathematics to test whether or not such a cut would actually give us more space on which we can slather our tasty cream cheese.

Calculating the Surface Area of Exposed Bagel in the Mobius Cut

In order to calculate the surface area of exposed bagel for our cream cheese, we will have to introduce some calculus: First, we will parametrize the torus using the following parametrization (in other words, re-representing the torus using a different set of variables):

The parametrization is (x,y,z) = P(a, θ,ϕ)=((acosθ+b)cosϕ, (acosθ+b)sinϕ, asinθ), where a is the distance of the point from the center of the circle that we rotate and b is the distance of the center of that circle to the center of the torus (which our toothpick passes through). The value θ tells us the angle between the segment connecting the point (x,y,z) and the center of the circle, and ϕ tells us how far we’ve rotated the toothpick (in other words, which disk cross section).

Second, we will consider the slice: a two-twist mobius strip. This slice includes all the points along a diameter of each circle. Furthermore, θ varies equally with ϕ, so we set up a parametrization for the mobius slice as follows:

(x,y,z) = M(r, θ) = ((rcosθ+b)cosθ, (rcosθ+b)sinθ, rsinθ) where a varies from -a to a and θ varies from 0 to 2π. After this, we will use a formula for finding the area of a parametrized surface, and we will find that the integral is quite the monster.

(For a more thorough explanation of the calculus behind parametrizing a mobius strip and calculating surface area, see https://www.mathstat.dal.ca/~antoniov/bagelrings.html).

The surface area of a straight cut through the center of bagel is more straightforward. If you’ve ever split open a bagel the normal way, you’ll likely have noticed that the exposed bagel is simply a circle with a hole in the middle. Therefore, we can find this surface area just by taking the area of the outer circle (π(a+b)^2) and subtracting the area of the smaller circle (π(a-b)^2).

Comparing the two areas using mathematical software, we find that the ratio of the surface area of the mobius cut to the surface area of the flat cut only changes when we change the ratio between a and b. Additionally, that ratio of surface areas is always greater than 1, so we can guarantee that the mobius strip will always give more surface area! However, the question of how much additional “bagelscape” we have for our cream cheese is dependent on the characteristics of the bagel shape. The greater a is in comparison to b, the greater the ratio of surface areas. That is, the fatter the bagel and the smaller the hole, the more benefit we get out of cutting our bagel using the mobius strip!

In theory, the shape of the bagel on the left will give less significant gains in surface area from using a mobius cut while the shape of the one on the right will give more gains out of using a mobius cut.

However, we run into a significant physical reality check when we realize that for extremely thick bagels, the cut will not work because we cannot fit the bagel through its own center hole. Therefore, we restrict

2a < 2(b-a)

2a < 2b – 2a

4a < 2b

a < b/2

Therefore, referring to our previous diagram of the torus, the Mobius cut will only work for a<b/2, in which case the surface area gain is sadly only about 5% from our mathematical software analysis.

Resources and Acknowledgements:

Hart, George W. “Mathematically Correct Breakfast: How to slice a bagel into two linked halves.” https://www.georgehart.com/bagel/bagel.html

Vargas, Antonio R. “On calculating a ‘Mathematically Correct Breakfast’.” (2014). Dalhousie University. https://www.mathstat.dal.ca/~antoniov/bagelrings.html

Weisstein, Eric W. “Torus.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Torus.html