#MathArtChallenge 93: Archimedean Solids from Vertex descriptions


The Challenge: Using a vertex description, build yourself one, two… up to all 13 of the Archimedean solids.

Materials needed: Card stock and tape (painter’s tape is great, or masking. Other stuff will work, but I’ve had more success with the paper-y tapes.) OR Magnatiles, but those can get pretty pricey.
Math Concepts: structure, polyhedra, angles, 3D structure

Platonic solids are 3D shapes with congruent regular faces. There are 5.

Archimedean solids are 3D shapes with regular (all congruent side lengths and angle measures) faces and identical vertices. There are 13.

So the idea is to get yourself a bunch of equilateral triangles, quadrilaterals, pentagons hexagons (and octagons or decagons if you’re feeling ambitious), and start building!

Below is a timelapse of me building the {3,6,6}, {3,4,4,4} and the {3,5,3,5}.

I think it’s fascinating how challenging it is to predict the finished sizes and number of faces. There are definitely ways to do it, though, so for students who want a challenge, see if they can figure out how many of each face they’ll need without looking it up or building it first.

Here are the vertex descriptions:

Platonic Solids

  • {3,3,3}
  • {4,4,4}
  • {5,5,5}
  • {3,3,3,3}
  • {3,3,3,3,3}

Archimedean Solids

  • {3,6,6}
  • {3,4,3,4}
  • {3,8,8}
  • {4,6,6}
  • {3,4,4,4}
  • {4,6,8}
  • {3,3,3,3,4}
  • {3,5,3,5}
  • {3,10,10}
  • {5,6,6}
  • {3,4,5,4}
  • {4,6,10}
  • {3,3,3,3,5}

To make it a bit easier on you, I have a sheet of 2 inch side length shape PDFs for you:

2in equilateral triangles

2in squares

2in pentagons

2in hexagons

One caveat to this vertex notation: the pseudo-rhombicuboctahedron:

This is actually an activity I have done with students before. Megan Schmidt and I got to run a summer camp for a week last summer and it was just glorious. It was so much fun watching students try to puzzle their way through making these shapes and then drawing connections between them. Some questions to consider:

  • How many faces or edges will each shape have?
  • How are the shapes related to each other? What connections do you see?
  • What other materials might you use?
  • Can you identify the shapes that are chiral? (They have a right or a left turning?)
  • Why are there only 13 Archimedean shapes? Why only 5 Platonic shapes? Can you find more? Why or why not?

Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

2.) Reason abstractly and quantitatively. Just based on the vertex description, can you figure out how many of each polygon is needed to complete the polyhedra?

7.) Look for and make use of structure. What combinations of polygons are possible? Which are not?

Author: Ms. P

Math Teacher in Minneapolis, MN.

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