PDFPRINT“3D Thinking” Page 131: An animation of the Pythagorean Theorem shown dynamically in terms of circles. In the animation the diameter of the outer circle 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 to the Theorem […]
PDFPRINT“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, […]
PDFPRINT“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” – but […]
PDFPRINT“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 primary […]
PDFPRINT“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 Mandelbrot […]
PDFPRINT“3D Thinking” Page 238: Shape Changing Polyhedra The original idea for shape changing polyhedra came from studying muqarnas and playing with children’s building blocks. The featured image is of the Nasir al-Mulk Mosque Muqarnas in Shiraz, Iran. The building blocks were as in the photo: An introduction to the logic of this geometry can be seen in […]
PDFPRINTThe concept originated studying the geometry of a particular window in old Cairo. “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 […]
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PDFPRINT“3D Thinking” Page 32 Neolithic Geometries: Labyrinths are constructed with a single path that winds forwards and backwards, multiple times, before arriving at a central point. The oldest surviving labyrinths appear as Neolithic and Bronze Age rock carvings – a petroglyph labyrinth in Usgalimol, India has been estimated to be 10,000 years old. By the time of the middle ages labyrinths […]
