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I never thought of that. Where did he say that?

by delish 10y ago

jacobolus 10y ago
Bucky Fuller was big on measuring shapes relative to triangular / tetrahedral coordinates. “Synergetics coordinates”. See e.g. http://www.rwgrayprojects.com/synergetics/s09/p6300.html#966.20 http://www.rwgrayprojects.com/synergetics/s09/p6300.html#966... The advantage of measuring area relative to a unit simplex is that it gets rid of a factor of n! in the denominator which would pop up if you measured the area of a simplex relative to a square grid. Likewise it simplifies the formulae for area/volume of the n–sphere. An equilateral triangle grid in 2-dimensions, and its dual the hexagonal grid, have much to recommend them. They’re often more efficient, e.g. it would be noticeably better (less isotropic, more efficient) to represent images using hexagonally shaped pixels than square pixels. Perhaps most importantly, they broaden thinking beyond the square-grid culture we’re immersed in in every aspect of our society (textiles, city planning, construction, carpentry, cartography, visual art, written documents, analytic geometry, ...) and just take for granted. It’s quite literally a way to “think outside the box”. The downside is that it’s not a priori obvious exactly which way to generalize the equilateral triangle grid beyond two dimensions, whereas for a square grid there’s a natural choice. We can base our coordinates on just a single simplex, but this doesn’t necessarily make measuring or relating arbitrary shapes any easier. The symmetry system of the tetrahedron has one type of 2-fold and two types of 3-fold axes of symmetry, and regular tetrahedra alone don’t tile space in a regular way. The FCC and BCC lattices, based on the symmetry of the cube/octahedron have 2-fold, 3-fold, and 4-fold axes of symmetry (this is the symmetry system of a tiling of space using mixed octahedra and tetrahedra). The dodecahedron/icosahedron have 2-fold, 3-fold, and 5-fold axes of symmetry, don’t tile space, and don’t generalize beyond the 4th dimensional case.
carapace 10y ago
I can't remember, and search failed (what's up with google these days? I search for "squaring" and all the results are for "square". :( ) but it was in one of his books, with a diagram. Jacobolus' comments are great.

I never thought of that. Where did he say that?

by delish 10y ago

I never thought of that. Where did he say that?

Bucky Fuller was big on measuring shapes relative to triangular / tetrahedral coordinates. “Synergetics coordinates”. See e.g. http://www.rwgrayprojects.com/synergetics/s09/p6300.html#966.20 http://www.rwgrayprojects.com/synergetics/s09/p6300.html#966... The advantage of measuring area relative to a unit simplex is that it gets rid of a factor of n! in the denominator which would pop up if you measured the area of a simplex relative to a square grid. Likewise it simplifies the formulae for area/volume of the n–sphere. An equilateral triangle grid in 2-dimensions, and its dual the hexagonal grid, have much to recommend them. They’re often more efficient, e.g. it would be noticeably better (less isotropic, more efficient) to represent images using hexagonally shaped pixels than square pixels. Perhaps most importantly, they broaden thinking beyond the square-grid culture we’re immersed in in every aspect of our society (textiles, city planning, construction, carpentry, cartography, visual art, written documents, analytic geometry, ...) and just take for granted. It’s quite literally a way to “think outside the box”. The downside is that it’s not a priori obvious exactly which way to generalize the equilateral triangle grid beyond two dimensions, whereas for a square grid there’s a natural choice. We can base our coordinates on just a single simplex, but this doesn’t necessarily make measuring or relating arbitrary shapes any easier. The symmetry system of the tetrahedron has one type of 2-fold and two types of 3-fold axes of symmetry, and regular tetrahedra alone don’t tile space in a regular way. The FCC and BCC lattices, based on the symmetry of the cube/octahedron have 2-fold, 3-fold, and 4-fold axes of symmetry (this is the symmetry system of a tiling of space using mixed octahedra and tetrahedra). The dodecahedron/icosahedron have 2-fold, 3-fold, and 5-fold axes of symmetry, don’t tile space, and don’t generalize beyond the 4th dimensional case.

carapace 10y ago

I can't remember, and search failed (what's up with google these days? I search for "squaring" and all the results are for "square". :( ) but it was in one of his books, with a diagram. Jacobolus' comments are great.