All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
A flat ball? (Posted on 2004-04-08) Difficulty: 2 of 5
Soccer balls are usually covered with a design based on regular pentagons and hexagons.

How many pentagons/hexagons MUST there be, and why?

See The Solution Submitted by Federico Kereki    
Rating: 3.7500 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Hint in the title? | Comment 6 of 24 |
I don't really understand what the question is asking: how many MUST there be?
I'll just add my thoughts to the fray.

The plane may be tiled with regular hexagons. They meet three to a corner at 120 degree angles.

If you were to replace some of the hexagons with pentagons but keep the sides joined, the figure would no longer remain flat. The way the hexagons are replaced is one at each vertex and no pentagons sharing an edge.
Each hexagon the borders 3 pentagons and each pentagon borders 5 hexagons. The ratio of hexagons to pentagons, then is 5:3.

Why does the soccer ball (truncated icosahedron) have 20 hexagons and 12 petagons?
Besides conforming to the above ratio and resultion fron the trucation of the 12 verticies of a 20 sided figure, I'd say one could look into the dihedral angles (but I'm not gonna).

One could ask:
Of all the polyhedra to choose from to make a ball, why this one?
-Lots of symmetry (all the arcimedian solids do)
-All sides nearly the same size (no dodecagons and triangles together, etc)
-Very close to a sphere (there are better, but it is really close)


  Posted by Jer on 2004-04-09 09:37:29
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2018 by Animus Pactum Consulting. All rights reserved. Privacy Information