Lattices
Contents
Two
Sphericons
Strings
Four
Sphericons
Nine
Sphericons
Blocks
of Nine Sphericons
Truncated
Octahedra made of Sphericons turns into a sheet
Cub-Octahedra
and Octahedra
Rhom
- cub - Octahedra and cub Octahedra and cube
Distorted
Great - Rhom - cub - Octahedra and Distorted Octagonal Prisms
Transformation
of Cubic Lattice to Truncated Octahedra Honeycomb
In a configuration whereby Sphericons are joined and poised to roll, they could be in a straight line, in closed rings of 4, 6, etc., spirals, and more. These strings could fold up into sheets, archimedean solids, etc. by only turning one a small amount.
Every half turn, two tangentially rolling pairs go out of synchronization
with the other two pairs by 1/18th of
a revolution (time wise). They go back in synchrony after a further 4/18ths
of
a revolution (time wise). This, in angular terms, is 45 degrees out of
synchrony. Therefore, 8/18ths
of
time, the mechanism is synchronized, a further 6/18ths
it
is unsynchronised, 2/18ths
it
is becoming unsynchronsed, and a further 2/18ths
it
is becoming synchronized.
This configuration can actually walk (as can be seen here). It always creates a flat surface on the base. When all four sphericons touch the ground, the configuaration can move in one of four directions. It can move back, forward, left or right. It can therefore walk in a square, zigzag etc.
Blocks of Nine will lattice together and will form fractal arrangements. Eight blocks of nine will roll around a central block of nine. This, we assume, will carry on for blocks of eighty-one.
Truncated Octahedra turns to Sheet
24
Sphericons will lattice into a truncated octahedra. You can put an infinite
amount of these truncated octahedra together to fill space. If you turn
one of these, the entire lattice will turn around each other and into a
sheet. Note that the Sphericons stay in blocks of four - these turn (see
four
sphericons). Click here
to see animation of this.
Look at the diagram above.
When the lattice is a truncated octahedra, all sphericons tangent (touch)
at c. When the lattice is a sheet, all sphericons tangent at a. This means
the sphericons , relative to each other, have only moved 1/8th of a revolution.
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Cub-Octahedra and Octahedra
These lattice together to fill space. Vertices (corners) of above shapes represent centre of each Sphericon. All Sphericons are poised to roll. If you look back at the diagram, in each cub-octahedron, the 12 sphericons tangent at b. In the octahedra, the 8 sphericons tangent at c.
The cub-octahedra (if made out of sphericons) butt together in the following way:
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Rhom
- cub - Octahedra and cub Octahedra and cube
These lattice together to fill space. Vertices of above shapes represent centre of each Sphericon. Again, all Sphericons are poised to roll. If you look back at the diagram, the sphericons tangent either at b or c.
The Rhom-cub octahedra looks like the following when made of Sphericons:
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Distorted
Great - Rhom - cub - Octahedra and Distorted Octagonal Prisms
To understand this lattice, it's best to start with the block of nine. If we simplify this we will use a shape where each vertices represents the centre of one of the outer 8 sphericons:
If you put 24 of these shapes together, you end up with a slightly distorted great-rhom-cub octahedron:
If you connect them using slightly distorted octagonal prisms, you can get an infinite lattice. This lattice can be modulised into a great-rhom-cub octahedron centred shape, surrounded by 6 octagonal prisms. This module represents 192 sphericons. If centre of blocks of nine are included this totals 216 Sphericons.
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Transformation
of Cubic Lattice to Truncated Octahedra Honeycomb
The lattice (above) falls about half-way along a transformation between a cubic lattice and a truncated octahedron lattice. Click here to see an animation of this.
For more information, contact: paul.roberts99@ntlworld.com
