# Various cross-sections of platonic solids

We're going to take the 5 platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) and suspend them in various ways (we'll assume that they are solid and of uniform density). Then we'll do a horizontal cut through the centre of gravity and describe the shape of the resulting cut face.

The suspension methods will be:

1. By a vertex (so any one of the vertices)
2. By a face (meaning by the middle of one of the faces)
3. By an edge (meaning the middle of one of the edges)

(This is a generalisation of a problem that I remember describing to a group, and only convincing them that the answer I gave was right by cutting up a potato. It was the cube, vertex case)

I've given some of the simpler answers in brackets, as examples. Bonus points for doing it all in your head...

1. Tetrahedron suspended by a vertex. (equilateral triangle)
2. Tetrahedron suspended by a face. (equilateral triangle)
3. Tetrahedron suspended by an edge.
4. Cube suspended by a vertex.
5. Cube suspended by a face. (square)
6. Cube suspended by an edge. (rectangle)
7. Octahedron suspended by a vertex. (square)
8. Octahedron suspended by a face.
9. Octahedron suspended by an edge.
10. Dodecahedron suspended by a vertex.
11. Dodecahedron suspended by a face.
12. Dodecahedron suspended by an edge.
13. Icosahedron suspended by a vertex.
14. Icosahedron suspended by a face.
15. Icosahedron suspended by an edge.

There's a couple in there that get a little tricky!! Enjoy!

• Hmm, with nothing restricting the answer, a general answer for all would be a closed polygon, wouldn't it? Still describing the shape ;c) – BmyGuest Aug 26 '15 at 6:32

• 3: square (cuts across all four faces symmetrically)
• 4: regular hexagon (cuts across all six faces symmetrically)
• 8: regular hexagon (cuts across six faces symmetrically)
• 9: rhombus (passes through two vertices and across two edges, cutting four faces in half)
• 10: regular hexagon (cuts across six faces symmetrically)
• 11: regular decagon (cuts across ten faces symmetrically)
• 12: irregular hexagon (passes along two edges and across two edges, cutting four faces in half)
• 13: regular decagon (cuts across ten faces symmetrically)
• 14: regular hexagon (cuts across six faces symmetrically)
• 15: irregular hexagon (passes along two edges and across two edges, cutting four faces in half)

The cross-sections for each polyhedron suspended from a vertex are the same as for its dual suspended from a face. The edge cross-sections are less regular because the orientation of the edge breaks rotational symmetries.

• Number 14 is not a hexagon. – Florian F Jul 7 at 7:30
1. Equilateral triangle
2. Equilateral triangle
3. Square
4. Regular hexagon
5. Square
6. Rectangle
7. Square
8. Regular hexagon
9. Parallelogram
10. Regular hexagon
11. Regular decagon
12. Regular hexagon
13. Regular decagon
14. Regular dodecagon
15. Oblique hexagon
• Any explanation??? – Rohcana Aug 26 '15 at 1:22
• Not all correct, I think. Not that I'm necessarily an authority – Dr Xorile Aug 26 '15 at 1:22
• I mean 10, not 9. Sorry. – Dr Xorile Aug 26 '15 at 2:03
• @Anachor I don't know what kind of explanation you are expecting. It's not a riddle or a strategy question, where you can explain your reasoning. For this question, that's just the way it is. No kind of explanation can change the answer or further clarify it. – Illyasviel Aug 26 '15 at 19:21
• Look at f" answer, I'm asking how you found those results, by working it your head, or by a software or by google, If you did it in your head, how did you visualise it, etc. – Rohcana Aug 27 '15 at 1:07