If you’re in the Seattle area, Studio Infinity and the Seattle Universal Math Museum will be hosting a freestyle build on Saturday, January 11th. We have absolutely no clue what this structure will look like when complete and would love for you to be part of the team that plans and executes its construction!
The unit we will be building with is a regular octahedral junction with eight triangular prismatic “arms” protruding from its faces:
We have a few ideas for what this building unit might be good for, but we are far more excited to see what a group of creative math enthusiasts can dream up.
The build will take place on Saturday, January 11th from 9:00am to 12:00pm in the 4th floor exhibit hall of the Seattle Convention Center. While the Joint Math Meetings have a registration fee the rest of the week, entry to the exhibit hall is completely free this Saturday. There will be many cool mathy things in the exhibit hall for both children and adults to enjoy, so bring friends and family of all ages!
In preparation for the announced JMM build, we worked out some kinks this past weekend with a practice build.
Since the final structure is just shy of 17 feet tall, it was important to have a venue with high ceilings. The Seattle Universal Math Museum has an ongoing partnership with the Georgetown Steam Plant, who generously offered their boiler room. (For fun, you can use their virtual tour to navigate yourself to the boiler room, as featured in the photos below.)
On Wednesday, 2025 January 8th at the Joint Mathematics Meetings in Seattle, Studio Infinity will be facilitating a group build using Giant Octablocks. If you follow our projects, you might recognize them as the trunctated octahedra from the TOWARD build at Harvey Mudd College or from a freestyle build at Open Sauce 2024. This time we will be leveraging the fact that the hexagonal faces of the blocks are at the perfect angles to build within a diamond cubic crystal structure if we restrict ourselves to a subset of four of them per block. The end goal — a Diamond Lattice Tower:
Our recent honeycomb lattice build at MathFest took up relatively little space for how rigid it was, so a natural question is what proportion of space the boxes take up in an infinitely repeating lattice. Since the boxes are themselves empty, a more amusing framing might be: how much of space would be boxed versus unboxed?
In our construction, we used boxes of dimension $a \times a \times b$. The way we thought about filling space with these boxes in their lattice arrangement was by complementing them with cuboctahedra with edges of length $a$ and irregular rhombicuboctahedron with $8$ equilateral triangular faces of edge length $a$, $6$ square faces of edge length $b$, and $12$ rectangular $a \times b$ faces:
From this perspective, the space occupied by these cuboctahedra and rhombicuboctahedra is the unboxed space.
To get a sense of how little space the boxes contain, we can look at the smallest cube containing our irregular rhombicuboctahedron. If we look at what else that cube contains from the honeycomb, we’ll see it’s $8$ eighth-cuboctahedra and $12$ quarter-boxes.
We know $V_{\textrm{box}} = a^2 b$ and can show that
$$V_{\textrm{cube}} = (\sqrt{2} a + b)^3$$
Since we can recreate the cantellated cubic honeycomb by arranging these cubes in a simple cubic honeycomb, the fraction of space taken up by the boxes is
$$\rho = \frac{3V_{\textrm{box}}}{V_{\textrm{cube}}} = \frac{3 a^2 b}{(\sqrt{2} a + b)^3}$$
When we have cubic boxes and $b = a$, this reduces to
A couple speakers dropping out at the last minute left a hole in the Math & Art session’s schedule the other week at MAA MathFest 2024 in Indianapolis, IN. As the famous expression you’ve no doubt heard countless times goes: when life gives you cancellations, make cantellations!
We arrived at Princeton early in the evening of August 5th with an assortment of the materials discussed in our planning post, ready to lead the PCMI/IAS Teacher Leadership Program in building an expanded icosidodecahedron.
Are you interested in bringing a mathematical art installation to your school or community? Studio Infinity would love to help!
Mathematical art has the power to inspire and educate. Installing a large work as a group helps foster or reinforce a sense of community around shared intellectual and artistic interests. And it can provide an opportunity for people to engage with and think about math in ways they may never have before.
With the PCMI/IAS Teacher Leadership Program build approaching, it’s time to pin down some details. We’ve used boxes to make a “cuboxtahedron” and wanted to work with a different symmetry, so we’ve selected the expanded icosidodecahedron as discussed at the end of this MathStream post:
For the PCMI/IAS Teacher Leadership Program, Studio Infinity will be proposing a build with boxes based on repeated application of John Conway’s ambo operator to a polyhedron, so we’ll take a brief look at that operator here.
After more or less independently stumbling upon the ideas for DodecaRT and TOWARD, a natural question is what polyhedra, in general, have skeletons that can be built using other polyhedra as vertex and edge components? Is there a way to generate families of new examples?
If you have access to a 3D printer and want a modular way to build TOWARD and other related structures, here are some options for putting together your own construction kit. And if you find yourself intrigued by the different ways these pieces can go together, you might enjoy exploring the mathematical questions they raise along with us.
