“3D THINKING” presents a history of the intimate relationships between culture, geometry, mathematics and design from the Neolithic period through the Indian, Egyptian, Babylonian, Chinese, Greek, Celtic, Islamic, pre-Columbian and Renaissance cultures, to the possible future.

3D Thinking in Design and Architecture from Antiquity to the Future”  Thames & Hudson Ltd., April 2018. Amazon USA, Amazon UK, Waterstones UK.

INDEX: Neolithic Geometries 32. The River Cultures 58. The Americas 106. The Pythagoreans 124. European Tribal Geometry 160. Geometries of the Early Islamic Period 178. The Renaissance 206. Out of the Past – Into the Future 220. Modelling with Equations 222. Fractals 230. Shape-Changers 236 12. Dynamic Circles and Spheres 264. The Future of 3D Geometry 294. Additional Thoughts and Ideas 312.

ANIMATIONS – A Supplement to “3D Thinking” by Chapter and by Page Number

NEOLITHIC GEOMETRIES – LABYRINTHS: Page 36 The Classic Labyrinth construction.


NEOLITHIC GEOMETRIES – LABYRINTHS: Pages 37 and 48. A Chartres Cathedral construction method using “switch” tiles.


THE RIVER CULTURES – ANCIENT EGYPT PYRAMIDS Page 78: Slope angles and relative sizes of nine pyramids of ancient Egypt


THE RIVER CULTURES – ANCIENT EGYPT PYRAMIDS Page 78: Slope angles and relative sizes of the Great Pyramids of Giza

THE RIVER CULTURES – ANCIENT EGYPT PYRAMIDS Page 78: Controlling slope angles during construction.











THE RIVER CULTURES – ANCIENT CHINA 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 hidden within which were the eight primary trigrams of the I-Ching. The following animation relates to the odd and even sequence on page 90 and a possible way by which Fu Hsi discovered the trigrams. The animation starts with the River Map then shows how the trigrams might have been derived followed by pathways of trigrams 0, 2, 4, 6 and then 1, 3, 5, 7. The last but one image in the animation shows the resulting order of trigrams and its symmetries – and the last shows the traditional arrangement.






GEOMETRIES OF THE EARLY ISLAMIC PERIOD: Page 190 Nesting Octagons. See the blog ‘An Octagon Design System.


FRACTALS: Page 233 The Mandelbrot Set. Black is within the set. Colors represent numbers close to being in the set. Credit lightscribe 1018389

DYNAMIC CIRCLES AND SPHERES: Page 271 Animation of the First Sequence


DYNAMIC CIRCLES AND SPHERES GEOMETRY: Page 274 Fig 12.30 Dynamic sphere geometry applied to 5-circles within the unit triangle of a square. The second close-packing is that used to generate the Altair Design lattices.


DYNAMIC CIRCLES AND SPHERES GEOMETRY Page 288 Lattices generated by connecting sphere centers – and structures abstracted from them.i











DYNAMIC CIRCLES AND SPHERES GEOMETRY: Page 293 Perceptual Patterns – Close-Packing Circle Lattice