“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


“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


“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

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

Geometry Architecture Golden Ratio Proportions

“3D Thinking” Pages 280 to 289: The drawing shows an example of how a Golden Ratio “Ideation Lattice” is derived from the “Dynamic Sphere” geometry – and then how an architectural structure is extracted from the lattice.  In this example, connecting sphere centers creates the lattice. The sphere packing is the “Golden Ratio” packing. SoftwareContinue reading Geometry Architecture Golden Ratio Proportions

Geometry Architecture Dynamic Sphere Lattices

“Ideation lattices” are three-dimensional latticed generated by the “Dynamic Sphere Geometry” to stimulate “visual logic” and creativity. They are the three-dimensional equivalent of “Perceptual Patterns.” Form extracted from an ideation lattice generated by the dynamic sphere geometry

Creative Geometry of Close Packing Spheres

I first developed the Dynamic Sphere Geometry in the late sixties; where spheres transform from one close-packing relationship to another, usually in finite sequences. The geometry intrigued me because most of the sphere arrangements generated had never been seen before and were unique. Since first presenting the Dynamic Sphere Geometry, in the early seventies, IContinue reading Creative Geometry of Close Packing Spheres