Altair Design – Page Update

Have updated the Altair design page with a new animation that shows the logic of how the designs are generated plus how the designs themselves can be perceived – with imagination, with logic, with creativity.

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Shape Changing Polyhedra – Updated Page

Have updated the Shape Changing Polyhedra page with movie clips and animations that introduce this new family of design and architectural structures.

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Dynamic Sphere Geometry – Updated Page

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.

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Bridges Conference 2016 University of Jyvaskyla, Finland

Just returned from the Bridges Conference held this year at the University of Jyvaskyla, Finland. Met some great people, mathematicians, architects, designers…

Presented “Shape Changing Polyhedra,” from my latest book “3D Thinking” to be published by Thames and Hudson Ltd Spring 2017. My presentation included stop frame animations and movie clips of shape-changing polyhedra full-shell modules connected in all sorts of ways so that when one part of an overall structure changes so do all parts.

My paper can be found at

The images below show the shape-changing polyhedra  “Shell #1 Module” and some combinations of the module. My bridges presentation featured seven different types of shell module.



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Radio Interview

Just appeared on KPCW; this morning in fact. Questions raised have made me think a bit more broadly about October’s, “Geomorph – Everything Changes When Mathematics Meets Art,” exhibition at the Leonardo in Salt Lake City.

I made a comment on-air that, when we teach mathematics and science, we should go back to the point of invention; to recapture the dynamics of an idea, and to give our students the opportunity to think about the mathematical, or scientific, concepts that we teach, as the inventions that they are. To think about whether the inventions are the best way of thinking, or whether there are better ways of thinking. Is the Earth flat, or is it more like a sphere, or maybe it’s like something else that is completely different? A lot depends upon what we’re trying to accomplish: to navigate across the ocean, or to try to figure out if there’s a gravitational benefit to launching a rocket from one part of the Earth rather than another. Re-captuturing the point of invention, of a mathematical, or scientific, idea, means taking a step back in time to the point of need, or to the point of discovery; to when, for example, we needed to find our way across the oceans, and the ideas explored to do it; and to look at the merits of one idea over another. If nothing else we might learn to appreciate the navigational mathematics that we have but, on the other hand, we might come up with something better, something more dynamic, something that better takes into account changing conditions, such as the wind, and the ocean currents. The Geomorph exhibition is all about approaching mathematics in a new way, and making it a medium of discovery.

I made another comment on KPCW that there are two types of mathematics, “Calculator Mathematics,” and “Discovery Mathematics.” Where schools mostly teach calculator mathematics. That we’re moving towards a time where computers can really take over the calculator side of things and that we, as humans, can get more involved in the creativeness of logical ideas – so that there is more of a partnership between us and the computers. Emphasis, then, in schools should, I think, be based more on logical problem solving and discovery. That’s not to say that we should forget calculator math but rather, re-position it. In fact, in my opinion, all students need to be familiar with calculator math, and we need some students who love calculator math, but also we need some students who question it; who question the underlying principles of computer programs; so that we never become over-reliant on mathematical models that might become obsolete, or limiting, or even destructive, as the the world evolves and new opportunities present themselves.

On the show I made another comment about how numbers don’t really exist but are just an abstract idea linking one lot of things with another; for example, matching a number of fingers on your hand with a number of sheep in your field. Mathematics is really all about abstracting ideas and making models of things. Once we ran out of fingers, to “count” things, we learnt to count in groups of ten. But that’s just the way our culture handled the problem of counting more than the number of fingers on our hands; some ancient Chinese counted in groups of two, Babylonians in groups of twelve, and Mayans in groups of five. There were many ancient cultures that never got beyond counting up to ten; they just reckoned they had, “lots.” There are so many abstract ideas floating around that, because we don’t question them, they take on a reality that they do not deserve. Money is a great example. Money is something we’re desperate to have;  that world commerce completely relies upon, but where the concept is really abstract; where the idea of “money” is, on the one hand, based on the real goods it can be traded for, such as food, water, and a roof over our heads; or on the virtual value based on what something might be valued at, if the equity worth was realized; such as the valuation of stocks, and currency. But where the value systems are highly variable and where one variable undervalues another, dependent upon the “credit” value or actual, “supply and demand.” Abstract ideas are very useful but also dangerous if we don’t evaluate and recognize them for what they are. Geomorph is all about the evolution of mathematical and geometrical ideas, but also about changing the rules and creating completely new things, new shape-changing structures and new geometries with applications in architecture, new materials, new structures, design, art, …

Hope to see you at the Geomorph exhibition!






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Have been working with four Park City, Utah, high school students on projects for Rockwell Collins and – as part of the PCCAPS program. The idea is to have students work on valid projects for corporations and to build experience with professional protocol, new types of software, problem solving, etc.

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Spheres Steiner Chain

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.

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Lectures: Geometry to enhance the human experience

My latest book, “3D Thinking,” to be published by Thames and Hudson Spring 2017 forms the basis for a new exhibition and lecture series, see below. Lectures feature photos, illustrations, movie clips, animations.

Please see my page, “Lectures and Exhibitions,” for descriptions.

Lectures include:

 The Golden Ratio – A Divine Proportion? Presented at the Leonardo Science and Technology Museum, Salt Lake City, March 2016.

The Future of Architecture. Presented at the Leonardo Science and Technology Museum, Salt Lake City, April, 2016.

Shape Changing Polyhedra. Presented at the Bridges Conference, University of Jyvaskyla, Finland August 2016.

Arabian Geometries.

The Pythagoreans.

A Dynamic Close-Packing Sphere Geometry. A unique new lattice generating geometry for architecture, design, science.

Labyrinths – Mysteries and Methods.

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Altair Designs Complete A Circle

For many years I’ve wondered where the original structure was that inspired the development of Altair designs.

Daud Sutton, a specialist in Islamic Design, sent me an email from Cairo just a few days ago. He has found found the original structure – a  latticed window – that has inspired so many of us – all that have explored Altair Designs, see below.

We can now complete the story of Altair:

A window’s lattice was constructed in about 1356 CE in the Mosque- Madrasa of Amir Salf al-din, Sargatmish in what is now, ‘Old-Cairo.’

Jules Bourgoin, in the early 1870’s, made a sketch of the window, created a schematic of it, and printed the schematic in his, “Les Elements de l’Arabe,” published in 1879 and numbered 151-  but gave no indication of the window’s location.

The design was then copied by Albert Calvert, without any mention of Bourgoin, in his, “Moorish Remains in Spain,” published in 1906.

Calvert’s book and design 151 found  its way to Dr.
Ensor Holiday’s hospital room in about 1968.

Ensor recognized the unique geometry of the design and started to create design variations. Close-packing circles were integral to the design and dovetailed my interest with Ensor’s – so we started working together – the rest is the story of Altair Designs.

So without that day in the early 1870’s, when a French architect sat down to make a drawing of a latticed window, a design that has inspired so many may have remained unknown and forgotten in the backstreets of an ancient town.

I wish Ensor Holiday and Aubrey Wolton (who helped place the designs with our first publisher, Longmans in the UK) were with us today – they would both be delighted to hear of the discovery.

Altair Window


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Think 3D to be published Spring 2017

Thames and Hudson will be the publisher! They will reformat the book and add their design and editorial magic. I cannot think of a better publisher! See my earlier posts re “Think 3D” and parts of my website to get an idea of the content. Will post more at a later date.

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