I started making these 'solids' during a rather intense migraine when i couldn't really focus on anything. i'd been thinking about interlocking spheres in the context of message exchange systems design - at least this was my excuse for spending hours making simple models of tetrahedra from triangular surfaces with semicircular flaps.
After i made a few of these and saw the platonic solids there, i remembered a passage from John Sowa's Knowledge Representation. Alfred Tarski used this approach to generate euclidean space using a sphere-based geometry which bypasses set theory.
take a line.
rotate that line around one of its ends.
then rotate it all the way around the end.
this describes two overlapping circles.
move the line so one end is at one of the two places where the circles touch. stick one end of it here and rotate it around.
now you have a nice surface, which i have come to think of as a 'tarski surface'. you can draw one and cut it out, or easier make one by taking three circles with the centre marked, and gluing them together according to the description above.
fold each circle in half and you have an equilateral triangle with semicircular flaps.
Now you can make all kinds of 'solids' or rather surfaces, both boring ones and weird ones! I see them more as extrusions - they are the equivalent of flat pictures of something three-dimensional.
Three circles overlap, their edge at each others centres, form an equilateral triangle. 4 of them glued together make a tetrahedron. 4 of *those* glued together with an equivalent terahedron of empty space inside, is illustrated by these pictures. the outward vertices of it describe a tetrahedron. it is made out of 48 circles. i wanted to see the interlocking spheres describing the whole thing.
The icosaholedron is really an icosahedron with two holes in its surface. It is nice to see the contents of an icosahedron. My favourite thing about this is that through it, you can see a hexagon. made this by accident while thinking about making a sexatetrahedron, and glued five surfaces together instead of six.
This slots into one of the 'concave' faces of the extrahedron. it's four tetrahedra (actually 2 tetrahedra both with one side missing) glued together, forming a pentagon shape on the plane, but with one 'slot' empty in the solid. Also by leaving one side off of each tetrahedron and save on paper circles.
Instead of making a 3-surface or a 5-surface with the original triple-circle-triangle shape, you can make a 4-surface and glue two of those together which produces a thing like an eight sided dice. So go ahead and draw the trigrams which compose the hexagrams of the i ching, all yin and all yang on 'opposite' surfaces, corresponding from there.
I look at this and think it should be possible to throw the i ching with this dice, treating the hexagram on each side like you would treat the three coins.
I must say that i don't especially like this structure. It doesn't have the integrity of the tetrahedron-based series. It "works", but it doesn't "feel right". If you take a flat plane at the same orientation as the halves glue together, you get a square.
I think of the three-circle pieces as 'tarski surfaces' now. In the same way as it only takes two circles to describe a triangle, the third one implicit, it only takes three spheres to define a tetrahedron, the fourth one implicit. This means you can save on paper by not having to glue all the surfaces together. These 'solids' are as much about the shape of the space they contain as the shape of the surface they describe.
It comes from this way of 'stacking' 3 spheres inside a sphere.
It comes from these two different ways of 'stacking' 3 spheres inside a sphere.
touching:
and interlocking:
You can see a really flat view of the extrahedron by sticking three tarski surfaces together:
Fabian thought that if a tetrahedron 'split open', a floating tarski surface could drift in and plug the gap, or perhaps it would just glue back together in a four-shape, not a three-shape, then be forever bound to find another four-shape to stick to, to make an i ching dice.
i'm not sure. i plan a 'tetrastore' instead of a triplestore for data, and am not sure about the design. i also am interested in designing a system of witnessed three-party transactions on this sort of basis.
you could play a nice kind of hexagonal game of go (would this work?)
One theory is that as an extrahedron 'travels' it describes a tube stacked with other tubes inside it. In some other space which we "can't see", it 'loops' in a moebius strip type manner which i find it hard to imagine. This picture i stole from mathworld, it is of something called a Cantor set:
That is a lengthwise cross-section of the stacked-circle-shape that the line appears in, if you extrude it into a stack of tubes, but with the centre circle "not there".
Thanks hex for taking the pictures and Fabian for talking it through.
2007-01-11