23 Mar

Summer 2021 PCMI Illustrating Math

This post was for announcing a week-long summer workshop on Illustrating Mathematics at the Park City Mathematics Institute, this past 2021 July 19-23. It was an exciting week with lots of interesting programming including several different hands-on, how-to tutorials, keynotes by Vernelle Noel, Ingrid Daubechies, and Daniel Piker, and mathematical “show-and-ask” sessions in which a wide array of mathematicians displayed some of the intriguing and beautiful images and objects they’ve created, as well as highlighting the questions these projects have raised. (I led one minicourse on using CAD/CAM software like LibreCAD or FreeCAD in creating physical mathematical models.)

12 Mar

Wirecosahedra

When I showed this recent post to my friend and colleague Laura Taalman, aka mathgrrl, she suggested that another approach to creating a model of the underlying structure would be to construct the icosahedra themselves (rather than the negative space), except use wireframes of the icosahedra rather than solid ones to avoid obscuring all of the internal structure. Her encouragement motivated me to create a new OpenSCAD file for this.

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09 Mar

Icosahedron in Octahedron

If you look again at the diagram of an icosahedron in a cube (at the right), you’ll see that because of its symmetry, all three of the icosahedron vertices nearest the top front cube vertex are the same distance from that cube vertex. That equidistance means that the body diagonal of the cube passes through the center of the icosahedron and is perpendicular to the nearest face of the icosahedron.

All of these properties hold at every vertex of the cube in relation to one of the eight faces of the icosahedron that do not have an edge lying in a face of the cube. Because the eight faces of the inscribed dual octahedron of the cube have these same properties, we’ve demonstrated that an icosahedron may also be inscribed in an octahedron like so:

It also means that if you extend the plane of one these particular eight faces of the icosahedron, it slices off an isosceles right triangular pyramid from the corner of the cube:

In a lattice of such cubes, eight of these pyramids fuse together at each vertex to create the octahedra seen inside the Anticos.

09 Mar

Icosahedron in Cube

As mentioned and illustrated in the post on the Anticos, it’s possible to inscribe an icosahedron in a cube. (In this case, that technically means that given a cube, you can choose two points on each face of the cube such that the convex hull of the resulting set of twelve points is a regular icosahedron.)

But why should this be so? To see this, it’s easiest to start with a regular dodecahedron, say with unit edge length. Notice the interesting pattern of the blue face diagonals in this diagram:

GIF

Notice that during this transformation, each of the edges remains in the plane perpendicular to that axis. Therefore, twelve of the vertices of an icosahedron lie on the surface of a cube, two on each face. Since an icosahedron only has twelve vertices, it is inscribed in a cube.

09 Mar

Anticos

Judging from at least one of the previous projects, Studio Infinity is intrigued with connecting polyhedra edge-to-edge. (Of course, connecting them face-to-face is interesting, too, but that’s pretty familiar from Legos and such; and vertex-to-vertex is the same as connecting dual polyhedra face-to-face.)

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