“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


“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: 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


“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” 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 labyrinthsContinue reading LABRYNTHS MYSTERIES and METHODS


“3D Thinking” Pages 22, 292: Patterns of Perception follow a methodology that has been applied through time from antiquity to the present. The first books to appear that feature “perceptual lattices” are books on designs developed during the early Islamic period – in the middle-east, north Africa, and in Moorish Spain. References are Jules BourgoinContinue reading PATTERNS OF PERCEPTION

Geometry Ancient Egypt – Pyramid Proportions

“3D Thinking” Page 74 to 87: The slope ratios, dimensions, construction, and alignment methods, of the Pyramids of Ancient Egypt: THE SLOPES OF THE PYRAMIDS OF EGYPT Given the whole number traditions of ancient Egypt one would expect the slopes of the pyramids to be in whole number proportions too. Whole number grids of squareContinue reading Geometry Ancient Egypt – Pyramid Proportions

A Geometry of Early Islam – Nesting Polygons

“3D Thinking” Page 190, 191: There are a number of Islamic design methodologies of which the “Nesting Polygon” method is one. The method places an arrangement of polygons within a polygon and continues infinitely placing the same but scaled arrangement within each polygon as it appears. The method was used to generate a lattice fromContinue reading A Geometry of Early Islam – Nesting Polygons

A Geometry of Early Islam – ABJAD, Circles and a Door

“3D Thinking” Pages 175 to 177 Islamic Geometry: This Seljuk period door dates from 13th-century Anatolia and stands as a possible example of a use of numeric values to communicate a message. The door’s design is based the Islamic “close-packing circle” method of design – in this case an arrangement of close-packing circles within a pentagon. The two primary numbersContinue reading A Geometry of Early Islam – ABJAD, Circles and a Door

A Geometry of Early Islam – The Dynamic Circle Method

“3D Thinking” Pages 187, 274: Using close-packing circles to create surface designs, during the early years of Islam, requires a method to generate different close-packing circle arrangements. The dynamic sphere geometry provides such a method. Applying the geometry, starting with a 5-circle arrangement within the unit triangle of a square, generates many arrangements including theContinue reading A Geometry of Early Islam – The Dynamic Circle Method