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LostInParadise's avatar

Can you visualize an icosahedron?

Asked by LostInParadise (27278points) 1 month ago

The icosahedron has to be the most neglected Platonic solid. The cube and tetrahedron are the most familiar. You have probably seen the dodecahedron with a calendar on it. An octahedron is just two Egyptian pyramids joined together at their bases.

With 20 faces and 30 edges, an icosahedron is quite an eyeful. I found this site to help visualize it. They could have done a better job of coloring, but if you stop the spinning and choose the beam option, it is not too hard to see what is going on.

Manipulate the shape so there is a top vertex and a bottom vertex,
each attached to 5 triangles. That accounts for 10 faces. Then there is a middle row, with 5 triangles attached at their base to the top row alternating with 5 triangles attached at their base to the bottom row. That makes for a total of 20.

Counting the edges can now also be done. The top vertex is attached to 5 edges. Each of the 5 triangles formed has a base edge. That is a total of 10 edges. The bottom vertex gives another 10 edges. The middle row has extra edges separating each of the 10 triangles. That adds up to 30.

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10 Answers

stanleybmanly's avatar

You apparently have lofty expectations regarding us.

lucillelucillelucille's avatar

That is an interesting link.
I imagine people that cut gems could easily picture this.

Yellowdog's avatar

I’m glad it’s platonic. I wouldn’t want anything more than friendship with it.

Very interesting in that it is something our brains cannot easily conceptualize.

Caravanfan's avatar

Sure. I play Dungeons and Dragons.

LostInParadise's avatar

It really is not that difficult to see it. I am not all that good at 3D visualization.
Maybe this works better:
https://commons.wikimedia.org/wiki/Category:Icosahedron#/media/File:Hawney's_complete_measurer-_or,_the_whole_art_of_measuring_Fleuron_T110822-18.png

Imagine that there are 5 triangles at G, even though you can only see three of them (GFB, GBC and GCH).

Attached to each of the bases of these triangles, there is an upside down triangle, like FEB and BAC. That is another 5 triangles.

Imagine that D is set up the same as G, with 5 triangles. The triangles attached to D are connected to the right-side up triangles like EBA in the middle row, for another 5 triangles, giving 20 altogether.

Jeruba's avatar

Ha. I can finally answer one (with something other than “no”).

My seventh-grade math teacher, much ahead of her time in various ways despite her white hair, had us construct them out of paper. We made dodecahedrons and tetra icosahedrons, too, and various other polyhedrons. And we did curve-stitching with colored thread in three dimensions. She introduced us to base 6, base 8, binary (base 2), and hexadecimal (base 16). We learned to do computations in bases other than base 10, on paper. No calculators or computers. This, my dear, was in the 1950s.

Math was never my subject, but I got A’s and B’s from her. I loved the way the abstract met the concrete in her vision, and the realization that base 10 isn’t natural and inevitable was a philosophical challenge that opened up vast realms for me. It turned out that a lot of givens weren’t actually indisputable at all.

LostInParadise's avatar

I am also interested in how the concrete and the abstract come together. Maybe it takes some actual construction to see how these shapes come together. I think you will agree that once you see it, it all makes good sense.

stanleybmanly's avatar

A gifted teacher can make an astonishing difference.

Jeruba's avatar

@stanleybmanly, I was lucky enough to have this brilliant teacher for three years in junior high school and a surpassingly outstanding English teacher in high school. Their influence affected my whole life.

@LostInParadise, we also made hexaflexagons.

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