' The Five Platonic Solids' ۞

An exciting look into the world of polyhedra, starting with the Platonic Solids.

The Platonic Solids are five polyhedrons with special properties, named after the Greek philosopher
Plato.

The faces of The Platonic Solids are congruent regular polygons, all meeting in the same
way around each vertex. The Platonic solids are isohedral, or face-transitive, meaning that they are made up of only one type of polygon each. They are isogonal, or vertex-transitive, meaning that each of their vertices are the same, and they are isotoxal, or edge-transitive, meaning their edges are the
same. They are also convex.

painting credit (Sarah McBeath) http://www.bakedbean.co.nz/platonics.htm

photography credit (golden models) http://www.flickr.com/photos/nomm/2307943186/

links

http://en.wikipedia.org/wiki/Polyhedron
http://mathworld.wolfram.com/Polyhedron.html

http://en.wikipedia.org/wiki/Platonic_solids
http://mathworld.wolfram.com/PlatonicSolid.html

http://www.tomlechner.com/sculptures/solids.html
http://www.georgehart.com/virtual-polyhedra/references.html

Platonic Solids and Plato
http://www.mathpages.com/HOME/kmath096.htm

Euclid http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/bookXIII.html

'It Was a Very Good Year' http://tmbw.net/wiki/Podcast_1A

Carl Sagan's 'Cosmos' http://www.youtube.com/watch?v=2NQg6W4SUaM

No comment: http://www.youtube.com/watch?v=xwq1yQiKmgM

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[CMrace]

Amazing at 3:50 I am a math teacher and I didn't know that. It's kind of funny they are duals and self duals. You think they would all be connected in one Geometric way. I also find it fascinating we find them so aesthetically pleasing.

LOL at the song! When he was imaginary?

My fav is the tetra always has been, but I like the isodeca also. Close second. I was never a fan of the 8 sided variety...something about the square in the middle bugs me. It feels like cheating...like a 10 sided dice.

[AdamLore]

@CMrace

It is really interesting to learn which shapes are dual to which and which are self dual. They are connected in one grand way, though. The more I learn about polyhedra, the more I see this. The dodecahedron, for example, is in many ways just a weird cube.

The octahedron is very much a "square" shape. The square pops up just as much in an octahedron as a triangle does in a cube. (or example, the vertex pattern).

Thanks for your comments!

[Mobrun2121]

i was in class when my teacher started playing this video, and there as quite a.... reaction from the song :D

[AdamLore]

@Mobrun2121

That is really funny!

[hannahtiz]

very useful thanks, but just wondering if you know what the minimum number of colours that would be needed to systematically colour the five platonic solids, with the rule that 2 faces with a common edge can't be the same colour. trying to revise for exam, can sort of work it out but want to make sure?

[AdamLore]

@hannahtiz

As far as I know there should be a minimum of four colors necessary.. I don't know too much about it, but my understanding is that the four color theorem applies to spherical surfaces (which would mean it applies to the Platonic solids.)

As a side note, The octahedron is unique in that it is the only one that can be colored with two alternating colors (like a checker board).

Hope that helps. Let me know if I got something wrong!

[RJL738]

Even without this, there are many possibly ways to work out the geometry of space. Theoreticly an infinite number of dimentions can be worked with, a trillion a Googol, Centillion tetrated to a Centillion tetrated to a Centillion to 10^303^10^303 levels, a vast amount of algorithms can be made. Geometry has many ecciting possibilities that reach beyond it's self.

[CovenantOfLove]

thank you for sharing. =)

[chriswilliamstool]

triangle is self dual :)

[RJL738]

@AdamLore Fair enough but yes the posibilities are wonderful. 

[AdamLore]

@RJL738

Thanks for the thought! I have a video on other geometric shapes as well. I think that in some way there must be an infinity of dimensions. As it turns out, though, beyond 4d, all dimensions only have 3 regular solids- the simplex, the hypercube, and it's dual the orthoplex.