“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 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
“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” 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
“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
“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
Have updated the “Dynamic Sphere Geometry” page. Applications are architectural, design, new materials. Revised page has updated animations showing how close packing sphere arrangements (of different sized spheres) are generated, examples of sequences and an example of an architectural lattice generated by the geometry.
In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles (blue and red) where n is finite and each circle in the chain is tangent to the previous and next circles in the chain.
Rethinking Randomness by Jeff Buzen introduces an alternative characterization of randomness and a new modeling framework that together explain the improbable success of these probabilistic models. Jeff used one of my close packing circle arrangements with a corresponding line lattice (connecting circle contacting points) and then an apparently random image selection (cover design) conjured by theContinue reading Rethinking Randomness by Jeff Buzen
We went to visit an old friend of mine, Nancy Stetson, in Boulder, Colorado, two weekends ago. About thirty-five years ago Nancy asked me if I would design an art studio, to be built on the foothills just above Boulder. This I did using one of the geometries that I had developed and that generatedContinue reading Nancy Stetson’s Art Studio, Boulder, Colorado