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.
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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.
How does the value of the following improper integral compare to 1? I.e., is it smaller, larger, or exactly equal to 1?
(This problem was proposed to Math Horizons Playground by Mehtaab Sawney of Commack High School. And for all of you $\pi$-ists out there, $\tau$ is of course just the radian measure of a full circle, i.e., $\tau=2\pi$.)
Stumped? You can peek at the solution using the password “strapyb”.
For definiteness, the unlabeled points do in fact trisect each of the sides of the triangle.
Stumped? You can peek at the answer using the password “threedian”.
What are the next few terms in this sequence of smallish numbers?
(And, of course, what’s the rule generating the sequence?)
Stumped? You can peek at the solution using the password “1p1a2s”.
A few days after the event at TCNJ, students at the PROMYS program at Boston University built another “Life sculpture” in which each layer is a generation and time proceeds downwards. Here, we explored questions of how you might know things like whether the resulting “sculpture” would be connected, or whether it would be self-supporting. For these types of questions, what one really needs is to solve the (more computationally thorny) “inverse Life” question: what colonies of cells can give rise to a given configuration in the next generation?
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Here is a photo of the first 13 generations of the evolution of the “R” pentomino pattern in John Conway’s Game of Life. Each layer represents one generation, and time proceeds downwards. In each layer, live cells are represented by boxes.
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Here’s an image from inside one of a pair of mirror-image snub dodecahedra built by passersby on the Harvard Science center plaza in 2019 April.
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Here’s a torus built from equilateral-triangle Geometiles that I used as a prop for an undergraduate talk at Harvard University in the Fall semester of 2018.
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