The prefix sesqui- comes from semi (half) + que (and, or also), so another half more or 3/2, and so the default meaning, rooted in that etymology, is surely just that, 3/2. The sesquitertia etc. smells to me like a back formation.

]]>I am thinking that if the n-ple vectors are a tile in a three dimensional space, then the n-ple cover the whole space, without the Bravais primitive vectors.

It could be possible to obtain each crystal structure using a closed path, and it could be possible to write each organic compound in a unique way (it is not the simplest, if there are many functional groups). ]]>

“Root lattice (maximal torus) of rank 8 of the E8 root system:

a Lie group of dimension 248, containing 240 root vectors spanning R8.”

Many years ago I tried to describe an “asteroid of revolution” (cusp fusion reactor)

to a blind girl using four pennies arranged in a square on a table and then telling her

to imagine rotating the central void around either the x or y axis. She seemed to enjoy

the puzzle. When it comes to these higher dimensions (space or group) I tend to go blind and

start thinking in terms of cube databases,

or dynamic videos evolving in space-time like Finnegan’s Wake.

Sam Loyd’s tangram paradoxes.

Frustrated protons? Why are clathrate hydrides hard to mine?

Thanks to my consideration of the cuboctahedron (skeletal) model i have made i feel i am better able to understand the concepts of face centred versus diamond cubic lattice structures – at least physically if not quite mathematically ;-) I saw where you were going with the red and blue lattices as being ‘different’, even though every atom in the structure is identical to every other atom, albeit with the orientation of their immediate four interconections being mirror reversed to that of all four neighbours. Is there not a mathematical symmetry in there somewhere, if not with the atoms themselves then how about around the point equidistant between any two atoms? Mirror symmetry?

Going back to the cuboctahedron it describes not only a sphere, and cube in 3D space it also consists of two mirror symmetrical (like the diamond lattice) tetrahedra. All four of which share a common central point. There is at the same time 6 identical pentahedra – square sided pyramids – with the same common point and oriented around the x,y and z axes of 3d space. This quite remarkable structure has revealed yet another feature (factor) to me today. It is relatively easy to see factors such as 0,1,2,3,4 and 6 as features in the cuboctagon, 0 being the centre or origin in 3d space, 1 being the basic unit from the centre to the periphery of the sphere or cube, 2 equal halves, 3 sided equilateral triangles on it’s surface or it’s 3 dimensional symmetry etc. but i could not see a way of factoring it by 5 before. (this was important to me because of my hobby of photographing Flora – 4 and 6 petalled flowers are common but there seem to me to be more 5 petalled ones. Thanks to the cuboctahedron and a close relative – the Rubik cube i finally was able to see how 5 and 10 can be factored into the picture.

This is in part thanks to your posts forcing me to think harder on the topic :-)

love.

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