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]]>In the Vedas it is stated: ‘He who desires heaven is to construct a fire altar in the form of a falcon. A fire altar in the form of a tortoise is to be constructed by one desiring to win the Hindu world of Brahman’ [the Hindu creative principle underlying the world]. There are traditions associated with each shape chosen – chariot wheel, Alaja Bird, Trough, circle, square, isosceles triangle, rhombus, etc.

Each of the fire altar arrangements shown has an area of 120 square units. When ‘squaring the circle’ Vedic approximations for Pi were close to 3.14. Concepts of “Squaring the Circle” may well have come down to us through the generations from Vedic traditions of building fire altars – beginning almost, 4,000 year ago within the Harappan, Indus Valley, culture, firstly by word of mouth and then later in the form of Sanskrit, and then through ancient Egypt, the Pythagoreans and Euclid…

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]]>An introduction the logic of this geometry can be seen in a paper I delivered to the Bridges Conference this August, 2016: http://archive.bridgesmathart.org/2016/bridges2016-225.pdf.

Shape-Changing Polyhedra are three-dimensional forms composed of polygons that are flexibly connected. Of most interest are shape-changing polyhedra ‘shells’ that connect in a modular fashion to fill space – to fill space and still retain their shape-changing characteristics.

Potential applications are endless and include things like packages that expand or contract to fit their contents; super tools that change their shape based on the needed function; robots and solar panels that fold up from a single sheet; shape changing aircraft, see shape transforming furniture, and transforming architectural forms. My work on shape changers is featured in my new book **“3D Thinking in Design and Architecture,”** to be published by Thames and HudsonApril 2018 and page references are to that book: Amazon USA, Amazon UK, Waterstones UK.

Core 1, Page 240 – Extended Shell 1

Core 1, Page 241 – Extended Shell 1 – Two Combined – Shows “Equilibrium” Positions. Equilibrium Positions are various stable positions / positions of balance for the shape changing polyhedra.

Core 1, Page 301 – Extended Shell 1 – Multiple Combined – Shows Various Equilibrium Positions

Core 2 Page 245 – Extended Shell 2

Core 2, Page 246 – Extended Shell – Multiple Combined

Core 3, Page 248 – Extended Shell 3

Core 12, Page 256 – Shell 12

See MIT origami robot on youtube

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]]>In two dimensional space images and patterns can be extracted from the lattices to form surface designs and perceptual patterns, see “Categories,” such as those used in the Altair Design and Images Design books. In three-dimensional space architectural structures can be extracted that will always be in proportion one with another; they can be modular, space-filling, and space-efficient. Stress loads and material usage is also easy to calculate, given the precision of the geometry.

The following animations compliment my latest book, **“3D Thinking in Design and Architecture – From Antiquity to the Future,”** to be published by Thames and Hudson Ltd, April 2018: Amazon USA, Amazon UK, Waterstones UK.

Animation 1, Page 271: An animation of a “first” sequence of seven close-packing sphere arrangements generated by the “Dynamic Close-Packing Sphere,” geometry:

Animation 2 , Page 284: The animation below shows the “Golden Rectangle Sphere Cluster,” that corresponds with the sixth cell of the “first” sequence, Animation 1, (Cell RTR 1.6) – where the cluster will tessellate infinitely in 3D space.

CLOSE-PACKING SPHERES: Close-Packing spheres are unique structures in that spheres touch each other and their centers triangulate. Close-packings of equal sized spheres have been known for millennia – to scientists, geometricians, and engineers, from Plato and DaVinci, to modern day chemists and molecular physicists. The manufacturers of almost anything circular or spherical use close packings to most efficiently pack their products, from nano-sized hollow spheres used in solar cells and lithium batteries, to packing tennis balls and optical cables. Structurally close-packing spheres and circles are stable and space efficient whilst their centers and contact points create infinite space-filling lattices. Molecules and atomic structures tend to follow close-packing arrangements and their lattices form many types of polyhedra. Their lattices make some of the most space efficient structures possible.

The “Dynamic Sphere Geometry” is based on algorithmic steps and defined parameters by which circles, or spheres, are allowed to change size and position within imposed limitations of symmetry and momentum. The parameters of the geometry can be changed, new limits imposed, new symmetries, non-symmetries, and the like, can be created to explore a broad range of sphere arrangements. The geometry can serve to “hunt”, indefinitely, for new and unique geometrical arrangements and for unique correspondences in three-dimensional space.

Animation 3, Page 272: Shows how the first sequence, Animation 1, tessellates across a two dimensional plane:

**Examples of structures generated from 3D lattices follow**, Page 288 – where the first is from a tessellating golden-ratio close-packing sphere arrangement and the second is from a tetrahedron-octahedron close-packing:

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