460: Torus

Posted by wonko on November 16, 2013 in geometry |

Christmas is just around the corner, so I was thinking “wreath” shapes and stumbled across an astonishing torus made entirely of Tom Hull’s “Phizz” units:

The structure is based on twisted units that combine in 5’s (a pentagon has positive curvature), 6’s (a hexagon has zero curvature) and 7’s (a heptagon has negative curvature).

The inside has 10 heptagons and hexagon spacers, the outer rim has 10 pentagons with hexagon spacers and the rest of the shapes are hexagons.

This shape does my head in – heptagons take up more paper yet less space in the shape … huh? Negative curvature makes the inside of the donut by making a series of “saddles” which is pretty neat.

I had 6 colours (about 555 units all up, wish I could remember counting them – I remember I also had a pile left over and used them for other projects), and mistakenly cut 4 squares per A4 sheet – mistake because the resultant unit ended up being waaaaay too big as I discovered later in the fold. I resolved to not repeat colours within any hexagon, to leave out a colour for each pentagon and to duplicate one colour on opposite edges of each heptagon.

To my amazement, eventually this OCD colour distribution worked (not without a lot of swearing, dismantling and re-arranging clashes) but it stands as a testament to persistence, lunacy and bloody mindedness.

There is a lot to like about this shape – if I had the patience to fold it again, I would do it much smaller and maybe monochrome because, quite honestly, mixing up the colours did my head in. “It could be a blue or an orange, but it cannot be orange because of the adjacent pentagon and it cannot be blue because of the blue next to it in the adjacent hexagon … so move the blue, swap it with a yellow, introduce a green and … aaaaarrrrrggggghhhhh!”

The Phizz unit is amazingly flexible, has two tabs but the possibility of 6 pockets+ meaning lots of different shapes are possible with it. If you remember I made a stellated icosahedron as well as my pastoral care group made a woven dodecahedron from them as well.

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