## FIRE ALTARS OF THE VEDAS – A LOGICAL PUZZLE FROM ANCIENT INDIA

“3D Thinking” Pages 67 to 72: Vedic Fire Altars present a logical puzzle for those building them. Firstly, the surface area has to be in direct proportion to the height of the individual whose task it is to construct the altar. Secondly, there is to be a precise alignment of the altar with the fourContinue reading FIRE ALTARS OF THE VEDAS – A LOGICAL PUZZLE FROM ANCIENT INDIA

## DYNAMIC SPHERES: GOLDEN RATIO ARCHITECTURE

“3D Thinking” Page 288: A “Golden Ratio” structure extracted from a 3D lattice generated by the sphere cluster, HP1.3. See Page 271, “3D THINKING ANIMATIONS.” The lattice is created by connecting the centers of the “Golden Ratio” spheres. The illustration shows how the lattice is generated and then how the  structure is extracted from the space-fillingContinue reading DYNAMIC SPHERES: GOLDEN RATIO ARCHITECTURE

## THE PYTHAGOREANS: PYTHAGOREAN TRIANGLES & CIRCLES

“3D Thinking” Page 131: An animation of the Pythagorean Theorem shown dynamically in terms of circles or spheres. In the animation the diameter of the outer circle/sphere equals the hypotenuse of the right-angled triangle. The diameters of the two inner circles/spheres equal the lengths of the other two sides of the right-angled triangle. According toContinue reading THE PYTHAGOREANS: PYTHAGOREAN TRIANGLES & CIRCLES

## VISUAL LOGIC: FIRST SIGNS – ICE AGE TO 3,000 BCE

“3D Thinking” Page 15. Evolution of signs towards symbols from 70,000 to 5,000 years ago – where the European ice age ended around 14,000 years ago. We see directional lines that criss-cross to form triangular shapes, shapes composed of directional lines that appear rectangular, linear arrangements of dots, ‘bell’ like enclosed shapes, feather like forms,Continue reading VISUAL LOGIC: FIRST SIGNS – ICE AGE TO 3,000 BCE

## VISUAL LOGIC: McGURK EFFECT

“3D Thinking” Page 22 The McGurk Effect: An example of how our brain forms perceptions from fragmentary sensory information. Here, from a BBC 2 program, Prof Lawrence Rosenblum pronounces the word “ba” continuously. The video shows his mouth pronouncing “ba” but then the video shows his mouth changing to pronounce the word “fa” – butContinue reading VISUAL LOGIC: McGURK EFFECT

## RIVER CULTURES: I-CHING

“3D Thinking” Pages 88 to 90 Fu Hsi’s He Tu (River Map): As legend has it Fu Hsi was fishing on the banks of the Yellow River some 5,000 years ago when a supernatural animal, a “Dragon Horse,” rose from the waters carrying on his back the river map within which were hidden the eight primaryContinue reading RIVER CULTURES: I-CHING

## FRACTALS: THE MANDELBROT SET

“3D Thinking” Page 233 Fractals: Named after the mathematician Benoit Mandelbrot. Fractals are defined as mathematical forms that are generated using a defined set of transformations that include a scaling function whereby the set will generate a form and then infinitely break it down into scaled repeats of itself or into, more generally, self-similar patterns of many types. The operational process for generating the MandelbrotContinue reading FRACTALS: THE MANDELBROT SET

## SHAPE CHANGING POLYHEDRA

“3D Thinking” Page 238: Shape Changing Polyhedra 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 fashionContinue reading SHAPE CHANGING POLYHEDRA

## DYNAMIC SPHERE GEOMETRY

“3D Thinking” Page 264: The Dynamic Sphere Geometry is a type of 2D and 3D logic that, in the first place, generates “close-packing” tessellating arrangements of circles and spheres that infinitely fill space. Connecting circle and sphere centers or adding clipped tangents or tangent planes, creates two- and three-dimensional lattices from which two- and three-dimensionalContinue reading DYNAMIC SPHERE GEOMETRY

## 3D THINKING PRE-ORDER FOR APRIL/MAY 2018

PRE-ORDER NOW! Available  Amazon USA, Amazon UK, Waterstones UK. and through other major distributors. Link for Book Description Second Description and Reviews