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For the Museum of Mathematics
MathMonday

Looking for just the right snack at your next “We Love Geometry!” or “Athenian Appreciation Day” party?  How about a regular tetrahedron of cheese?CheddarTetra

How did the Math Mondays Food Labs slice this simplex? The first step is to select a nice, semi-hard cheese: cheddar works nicely, for example, but avoid the extremes of goat or Parmigiano-Reggiano. Make sure to select a sharp knife, and begin by cutting as near-perfect a cube of cheese as you can. Note that the final tetrahedron will stand 2 / √3 times as tall as your cube (HW: prove this), or about 15 percent taller than the cube, so keep that in mind when choosing the size of your cube. Here’s what Math Mondays started with: CheddarCube

Now, choose any face of the cube, and with the sharp edge of your knife, lightly score the diagonal of the face. On an adjacent face, lightly score the diagonal that connects at one corner (call it corner A) with the first diagonal you made. Place the cheese cube on a cutting board with corner A on top pointing toward your cutting hand (i.e., toward the right for me, as I’m a righty). The corner directly below A we will call corner B. Now slice from corner A diagonally down away from your cutting hand, carefully making sure to keep the two spots where the knife protrudes from the cheese in front and back travelling along the two diagonals you marked.

FirstSliceStraight  FirstSliceSide

Slice all the way through the cheese on this plane, until your knife simultaneously reaches the two corners diagonally across those faces from A. This operation cuts off corner B along with a pyramid of cheese, and leaves an equilateral triangle cross section on the main block of cheese you’re working with.

OneSliceDone

Now you’re really home free: you are just going to repeat this process three more times, cutting off all three corners of the cube which are not adjacent to B but are adjacent to a corner adjacent to B. Note that on the second cut you will be cutting along one of the edges you created with the first cut and simultaneously along a newly-marked diagonal, and for the third and fourth cuts, you will simply be cutting along two newly-created edges. The tetrahedron is already somewhat recognizable after the second cut:

TwoSlicesDone

And it is very clear after the third:

ThreeSlicesDone

And when you have made the fourth cut, you are left with a regular tetrahedron, and four identical “isosceles” tetrahedra that have three identical sides:

FinishedCheddarTetra

Happy simplex snacking from Math Mondays!  Feel free to send pics of other edible geometrical creations to mondays@momath.org as possible fodder for future columns — or just fodder for MoMath staff!

Glen Whitney

Executive Director, Museum of Mathematics


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Comments

  1. Joel Finkle says:

    Getting the octahedron isn’t much tougher. Draw X’s on the top and bottom faces, and draw an equator around the middle. Cut each side from the center top to the equator — eight cuts gets you the octo, with the first couple slicing off larger chunks than the later ones.

    1. Keith Neufeld says:

      That won’t give you a regular octohedron, though. To do that, cut all the corners off the cube, so that the square connecting the centers of the sides of the cube becomes the “equator” of the octohedron.

      A polyhedron formed by connecting the centers of the faces of another polyhedron is called its dual. Tetrahedra are their own duals — connect the centers of the faces of a regular tetrahedron and you’ll have another tetrahedron (4 faces and 4 vertices). The octohedron (8 faces and 6 vertices) is the dual of the cube (6 faces and 8 vertices) and the icosahedron (20 faces and 12 vertices) is the dual of the dodecahedron (12 faces and 20 vertices).

  2. If the Scalene Triangle is a symbol of Love, the Isosceles Tetrahedron is ?