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Here are some things my high-school students have made....

At the Mathematics Triple 2012, we had a polytope construction session using Zometool:

Sarah Liebowitz made tetrahedra inscribed in two cubes of a hypercube.

Matt Gu holds aloft a 5-cube.

Bradley Zykowski added struts in every possible place to a Zome ball, and in doing so created an icosahedron within a dodecahedron.

Okay, so technically these cookies were not made by high-school students. They were made by math graduate students at a tessellation cookie-making party over Thanksgiving in 2013. But in summer 2006, these folks were in high school and were my students (photographic evidence of this is below)...

Nate Harman made and ate an icookiehedron (or icosacookie).

Hannah Alpert grins near her Sierpinski cookie.

Gabe Karpman knitted a projective plane. How cool is that? (at HCSSiM 2007)

HCSSiM 2005 had a doughnut-making party. Below are before-frying and after-frying photos of the Sierpinski Triangle doughnut, made (I believe) by Andy Feldman. Note also the lame triple-holed torus (proabably made by me).

No doughnut-making party is complete without frosting doughnuts, and what better frosting pattern than a graph embedding? Here Noah Blach shows off his embedding of K_6 on the two-holed torus.

My Polytopes and More maxi class at HCSSiM 2006 was trying to decide on a definition for regular polyhedra. As part of their investigations, they invented some infinite polyhedra (Coxeter-Petrie polyhedra, in fact)...

June Jhe with his 6-squares-around-each-vertex creation

Nate Harman with his 4-hexagons-around-each-vertex and 6-hexagons-around-each-vertex models

Not to be outdone, Emma Cohen and Hannah Alpert constructed an 8-triangles-around-each vertex polyhedron, and then concluded that it couldn't be regular.

Emily Bargar, Susie Ko, and Max Engelstein generalized Pascal's Triangle to Pascal's Tetrahedron. Here they are at HCSSiM 2004 creating a model of it from packing peanuts, marker, and toothpicks.

Jacob Kushkuley (HCSSiM 2004) made a really large hyperbolic hat (named ZeZe by Emily Bargar, pictured earlier) with 8 triangles around each vertex. What is difficult to see in these photos is that he had attempted to create a closed surface by attaching free edges wherever possible and thus creating doughnut-like holes. In other words, this partial embedding is of very high genus.

In the following five pictures, you see the result of string pieces choreographed by the Dr. Schaffer and Mr. Stern Dance Ensemble. (More detail on their pedagogical work is at the Math Dance site.) These took place at the Hampshire College Summer Studies in Mathematics (HCSSiM).

Clockwise, beginning with the top: Andrew Uzzell, Karol Przbytkowski (seated), Matt Tretin, Daniel Studenmund, Martin Vieth (center), Gavin Dow, Danny Scheinerman, Niko Warner, Nelson Chiu, Maria Guirguis. Lying in the background is Leathan Graves-Highsmith.
This is a dodecahedron from multiple string loops, from HCSSiM 2003. The students used diagrams and instructions I wrote from discussions with Karl Schaffer.

Clockwise: Emily Bargar in green, Ian MacLeod, Jamie Conway, Sho Uemura, Talia Hurwich, Alex Kunin, from HCSSiM 2004. In the left-hand photo, the students are discussing how to transform an octahedron into a tetrahedron, which has been completed in the right-hand photo.

Clockwise from left: Jan Siwanowicz (HCSSiM 1993), with Azi Horowitz, Jacob Izraelevitz, Alex Bowen (visitor), Michael Gottlieb, Max Woerner-Chase, Richard Bowen, ???, Noah Blach, and Aaron Wilkowski, from HCSSiM 2005. This is a single-strand dodecahedron.

Left to Right: Amanda Redlich, John Choi, nilmA ffeJ, Albert Wang, Sergei Lupashin, Emily Riehl, Ian Wang
This icosahedron is from HCSSiM 2000. The students were given the original Schaffer/Stern instructions and directed by me and then by my junior staff Matt Riddle.

My topological graph theory maxi (HCSSiM 2004) suddenly decided to form a human Petersen graph. Each person is a vertex, extending two arms and a leg representing the three edges emanating from each vertex. In the right-hand photo we'd decided it would be a good idea to actually form a connected graph; too bad the photographer's finger got a bit in the way.
Outer pentagon, clockwise: Jon Strassfeld wearing his yellow pig shirt, Brendan Foley, Jonah Blasiak (junior staff), Rachel Kirsch, and David Lonoff.
Inner penta-star, clockwise (not by edge-adjacency, but by the usual people-adjacency): me in my yellow pig dress, Hannah Alpert, Sho Uemura, Alex Kunin, and Bret Wilson.

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