3D Thinking in Design & Architecture – April 2018

3D Thinking – Geometry culture and design throughout human history – April 2018

Will be available internationally through Amazon, Thames & Hudson UK and USA. UKP 39.95. 328 pages, over one thousand illustrations and photos.

This unique and inspirational design resource presents a history of the intimate relationships between geometry, culture and design throughout human history, from the Neolithic period through the Indian, Egyptian, Babylonian, Chinese, Greek, Celtic, Islamic, pre-Columbian and Renaissance cultures, to the present and the possible future.

Explains key principles that can be applied across all design disciplines, Reveals fresh insights into how geometry as a visual language has evolved to meet our needs, initiated new technologies, solved problems and changed the way we think about the world around us.

  • An essential sourcebook for design and architecture students, professionals, and general interest readers.
  • Covers humankind’s approaches to design and architecture across the entire range of human history and culture.
  • Stimulates new ways of thinking about the pressing design challenges of our present and future.
  • Illustrated with over one thousand specially created diagrams and artworks.



Have been working with three high school seniors, and one junior, on projects for Rockwell Collins and Overstock.Com – as part of the PCAPS program, Park City, Utah. The CAPS concept is to link students with corporations and have them work on real projects that will build their professional skills, broaden their experience with hardware and software platforms, and push them into situations that need innovation and sheer hard work. The learning curve for some of the software is pretty steep – but the students are doing well. Next semester I’ll have ten students and we have some interesting projects lined up including one for the USAF.


The publisher has asked for new chapters and I have been in the process of adding chapters that cover the geometries of the “river cultures,” – Indus Vally; The Nile; The Euphrates; The Yellow River. I have also broadened the Celtic chapter content to include a broader range of European tribal geometries – primitive as they often are. Also a new chapter on the European Renaissance geometries. To develop the new sections and chapters I have had to do lots of research but the new material really helps show how visual concepts have developed over time, how local cultures have impacted concepts, and also how unique many of the concepts were, and still are – and how these unique concepts of the past can be applied in the present and possible future.

The page count will probably end up around 350 bt could reach 400 – full of illustrations, diagrams, designs, and links to animations, and movie clips. Throughout the book I’ve worked hard to be original and within the pages you’ll find many new discoveries and insights. I have printed samples of the book, “working copies still in process,” for review.

Geomorph PCTV interview

Links for last weeks PCTV interview, “Geomorph – Everything Changes When Mathematics meets Art”:

Part 1 http://www.youtube.com/watch?v=RZBLr-qUrqo

Part 2 http://www.youtube.com/watch?v=vnFfnExZNZo

Think that my “jargon” is slightly better in the KPCW interview on http://kpcw.org/2012/09/the-mountain-life-sept-12-2012/

Hope to see you next month at the Leonardo!

KPCW – Talking about Geomorph and about the process of invention

Appeared on KPCW this morning to talk about my upcoming exhibition, workshops, and residency, at the Leonardo, in Salt lake City – Utah’s Creative Science and Technology Museum. If you’d like to hear the broadcast please visit: http://kpcw.org/2012/09/the-mountain-life-sept-12-2012/.

We talked about my experience as an inventor, and what it takes to be one, but, primarily, we talked about the Geomorph exhibition and how it concentrates on “Discovery Mathematics,” rather than on “Calculator Mathematics;” and about how we need to go back to the point of discovery, to the point in time when a mathematical idea was created, to really understand it, and then, maybe, to change it, or maybe to come up with something better, or more versatile…

Hope to see you at the Leonardo during October. The exhibition starts on October 3rd and I’ll be in residence for the first two weeks and hold organized workshops on the 5th, 6th, 7th, 12th, 13th, and 14th at 3.30pm – but otherwise I’ll be happy to talk and to share.

The Geomorph exhibition runs concurrently with the exhibition, “DaVinci The Genius,” which opens on September 28th and runs through to January – so come to both in October!

The Leonardo: Everything Changes When Math Meets Art

The “Geomorph,” exhibition date approaches and I’ll be building shape-changing geometrical structures; preparing large high definition LCDs, to display animations of new and ancient geometries, setting up an interactive program designed as a visual and geometrical game of logic; printing out posters that show the geometries and the mathematics behind them, and collecting materials so that visitors can build and discover new 3D forms of their own, etc. The Geomorph theme builds on the concept of making geometry a medium of discovery and on the idea that there are two types of mathematics, “Calculator Math,” and “Discovery Math,” – where schools tend to teach, “calculator math,” and thereby miss the whole idea of “discovery math,”  where students can really be innovative and explore new ideas.

To get an idea of the types of geometries that will be shown please see the page, “Mathematics Meets Art Exhibition.”

Curiosity Rover Lands On Mars

Congratulations to JPL, and the whole NASA team, on successfully landing the Curiosity Rover on Mars; and thanks for sharing the landing experience on NASA TV last night! Curiosity travelled from the Earth to Mars in about eight months, traveling 352 million miles (569 million km), leaving a rotating and orbiting Earth, to rendezvous with Mars, also rotating and orbiting the Sun, and landed within the target site in Gale Crater: very cool and precise calculations, but also an amazing feat of engineering. On entering the atmosphere of Mars, at some 13,200 miles per hour (21,000 kph), the space craft slowed to zero using a heat shield, a supersonic parachute, and a rocket powered sky crane to gently lower the one ton Curiosity Rover onto the surface of Mars. Very, very, cool. So now we can look forward to exploring Gale Crater and the three-mile high (5 km)  Mount Sharp that rises from the crater’s center. Maybe it’s tough to justify the US$2.5 billion spent, but the Curiosity mission is inspiring, creates a new technology knowledge base and, hopefully, will provide more insights into why Mars is such a dry planet, where the water was and still is, and, maybe, discover that life did exist, or still exists! Curiosity adds one more building block to a possible future of humans living on Mars and developing new opportunities for mankind using the planets resources and low gravity environment . Check out the NASA site: http://www.nasa.gov/mission_pages/msl/index.html For a simplified way to calculate a path to Mars check out: http://www-istp.gsfc.nasa.gov/stargaze/Smars1.htm









Image credit: NASA/JPL-Caltech

(i) Leaving Earth. (ii) Supersonic Parachute deployed and craft moving to Gale Crater on right. (iii) View of Mt Sharp (3.4 miles high) after landing. (iv) Earth and Mars elliptical orbits, both planet’s rotating about their own axis. Flight path is calculated to take in speed of space craft, orbits and axial rotation of planets, Mars atmosphere, and means to slow the space craft to a stop. (v) The base of Mars’ Mount Sharp – the rover’s eventual science destination – is pictured in this August 27, 2012 NASA handout photo taken by the Curiosity rover.  (vi)View of rim of Gale Crater. All NASA and JPL photos.

Curiosity landed facing east-southeast within Gale Crater, with a heading of 112.7 degrees (plus or minus five degrees), and a few degrees of tilt. A Sol 1 overpass by Mars Odyssey will provide additional information on Curiosity’s position and additional imagery.

The Leonardo, Salt Lake City. When Math Meets Art Everything Changes

Do you want to think about geometry in a whole new way? Explore ideas and create amazing designs and dynamic 3D structures? I’ll be presenting old and new geometries that create everything from labyrinths to moon bases at The Leonardo in Salt Lake City, Utah, during October, 2012. The presentation will take the form of a small exhibition/workshop of high-definition animations, 3D structures that change shape and size, geometrical designs that change before your eyes, and all sorts of models and diagrams. The exhibition/workshops falls under the name, “Geomorph – Everything Changes When Math Meets Art.” I’m hoping that there will be funds, at a later date, to put on a full exhibition with lots of interactives – structures you can walk into and have them change shape all around you, etc. For October, and at the moment, there’s a tiny budget, but still, we’ll do the best we can! If we can find corporate sponsors then we could really make an impact on transforming math from a sort of calculating machine into an adventure.


In my opinion most of us have the wrong opinion about math. Even saying the word, “math,” sends shudders down the spines of most people – and many people will blank out immediately they hear the word! So, for those that have managed to read past the dreaded word, I’d like to say the following…

Things such as Algebra and Calculus have been pushed upon us and they are called Mathematics. Trouble is that they are mathematics, in as much as such things are the tools and means to calculate. But the heart of mathematics is not such things as calculus, algebra, and trigonometry, it is, rather, a way of thinking.

When we think about things, analyze them, think logically, and wonder how things might work; or model something to try to understand it; we’re acting as true mathematicians. In fact we can be hopeless at algebra but still be great mathematicians*. In my opinion most of us have been, “lead down the garden path,” so to speak. We’re taught so-called mathematics in school, in the hope that we’ll acquire the various tools of math but, mostly, we have no idea why we have to plod through the drudgery! Confusion reigns!

Take a step back and ask yourselves why and when you need to think logically, analyze a situation, figure things out. “Most of the time!” right? So where’s the disconnect?

School mathematics is largely just a “tool,” sort of like a calculator. But it’s a calculator without a store/shop, so to speak – we’re shown little purpose for it, there are, for example, no prices to add – and I don’t mean the math of money. The “need” for the tools of math is sidelined – so too is true math! We’re just taught the tools with little or no application except for, maybe, money, and something to do with angles and lengths. Now clear your heads for a moment. Forget the word “math,” and instead think of an idea you’d like to explore, something to build, make, model… Now think of the elements of your idea, break them down into components, then see how they combine. Play with the combinations, look for originality, dare to be different, be unafraid of mistakes. Then, maybe, see what sort of math “tools” you might need – and it might not be anymore than a flow chart, measuring cup, compass or ruler..but it could be trigonometry or calculus (oh, oh!).

Geomorph is all about thinking, analyzing, and logic. It looks at the logical geometrical systems of the past, such as the logic of building a labyrinth 4,000 years ago. It also looks at geometrical systems of the future. In a way geometry is the visual language of mathematics. You can use the simple or complex tools of mathematics in geometry, but you don’t have to. The exhibition aims to show things that are surprising – things that you have never seen before. Its math that moves, transforms, changes – and it’s dynamic, and serves as a means to discover. With Geomorph you’re no longer a math calculating machine but an explorer!

A key thing to note is that, at one time, no mathematical concept existed. Every branch of mathematics was, at some point, invented, usually in response to a need: “We’re lost, how do we find out where we are?” “How tall is that mountain?” “Counting on our fingers is OK but how do we count really big numbers?” “How do we figure out how to send a spacecraft to the moon?” etc. It’s insightful to remember that there was always an initial need  or a problem to be solved, and a beginning, to each branch of mathematics. If we can go back to, “the point of need,” and to the, “beginning of each branch of mathematics,” we can not only rekindle the excitement, and drive, that lay behind the original discovery or solution, but also, better understand the math, and why it was or is useful, how it can apply, or, even better, figure out a better way solve the original problem. Focusing on the need might lead to novel and more efficient solutions as an alternative to math languages that have grown in complexity as they have evolved. It’s a bit like programming and using a lot of old code, “cut and pasted,” into an application, or operating system. Maybe people don’t even think about the efficiency of that old code because it’s so deep in the pile  – but maybe it’s time to take a new look at those old routines, because they might be too complex, or not so efficient, or even, illogical.

* According to my old friend and mentor, Dr. Ensor Holiday, Albert Einstein, travelled with his own personal, “mathematician.” Not sure if this was Marcel Grossmann, or not, as I cannot remember the details of Ensor’s meeting with Einstein.  From my perspective, Albert Einstein was a “discoverer” mathematician, and Grossmann, if Grossman, served more as a “calculator” mathematician – at least as far as Einstein was concerned. I stand to be corrected if anyone knows more…

Young Inventors Summer 2012








Just completed two summer workshops at the Kimball Art Center, in Park City, Utah. The morning sessions were for “Teen Inventors,” aged from 11 to 17 years, and the afternoon sessions were for “Young Inventors,” aged 5 to 10 years. Most inventors completed an invention a day – which was pretty fast going! One vehicle had a phenomenal speed – probably because the motor was loaded to the max with a much higher than spec. voltage – the gearing was pretty aggressive too. There were two attempts at building types of hovercraft but more work was needed to improve performance – problems were with an air flow-back. Performance, oddly, could well have been improved by allowing air leakage but we ran out of time. We also had two water based vehicles and all sorts of flying and land based vehicles.

We tried different ways to add a new dimension to the workshops by adding game elements – racing, distance, point scoring in game play etc., but the most successful was a sort of “battle pit” where insect like mini-bots interacted in a confined walled space. See the video by pressing the first photo above. Next time I think we’ll spend time building walled tracks with obstacles – to give more of a challenge to the land based inventions.

Visual Imagination and Invention

This web and blog site is primarily about geometry and about new ways to model structures in space. But it is also about imagination and the need to be flexible with the “models” we create, whether they are geometric, or not; modeling anything from an atom to the way a rain drop falls, or how the weather changes.

Visual imagination comes in many forms. Sometimes there is a type of image triggering in our minds when we look at cracks on a wall, clouds in the sky, rock formations, or the twisted bark of a tree. This type of triggering is often unconscious and can surprise, or even frighten us; when suddenly we see a figure in a tree under a moonlit sky, only to find that it is just the tree.












We often complete images in our visual imaginations with way too little information. This can happen accidentally, or when we’re trying to visually understand something. For example, a rock might appear to be a sleeping man, and the image might be really persistent in our imagination, but then we get more visual clues, and we realize that it is just a rock. We do the same sort of thing when we think we recognize someone in the street, but then find we are wrong. Sometimes, when we complete a visual image with insufficient information, the image stays with us, and we find it hard to give it up, even if it is an incorrect image. A good example of this was the misunderstanding that there were canals on Mars.








A reverse process occurs when we have a mindset for how something is, such as the Earth being a sphere, or an atom looking like a mini-solar-system. The mindsets are only models that we create, and can be very useful, but also limiting. Models of an atom range from that of Democritus, 400 B.C. to quantum models and energy profiles. Each model serves its purpose but none of them capture all of the apparent properties of an atom.





If our models are too strong, or have been overly reinforced by our teachers, then we can become blind to what might really be going on, or to new opportunities or possibilities. We could have a debate as to whether the Earth could ever have been a sphere, and under what circumstances it could find this state of mathematical perfection. It’s only when we realize that the Earth is not a sphere that we can start to understand other things that are going on.

Sometimes misunderstanding what we see leads to new ideas that might apply to something else. Other times, misunderstanding something puts us on alert, when we do not have enough visual information, and we may be right to be wary. But, on the whole, in our relatively safe world, with few predators around, we need to be careful with our imaginations; sometime let them run free, but other times direct them, or question their validity. We have no real, all inclusive, mindset for quantum physics at the moment, even though many are trying to create models to help us understand what is going on – but, on the other hand, it’s wonderful that we do not have any set models for quantum effects; because possibilities remain open. The same is true for many of established mind sets about gravity, about the internal combustion engine, about almost everything – our models can inhibit our thinking. Our visual imaginations help us survive and allow us to do wonderful new things, but sometimes we are mislead by them, and they breed prejudicial ideas. It’s just a matter of how we direct our visual imaginations. We need to be aware of the dangers of the beast!

I’ve always liked the Johnny Hart, BC, comic strip that shows an improvement to the concept of a wheel; from a square wheel to a triangular one. Conceptually the mind set is going the wrong way. Instead of removing a bump, the better solution would be to add bumps, lots of them, in fact an infinity of bumps. Eliminating bumps might have worked if conceptually one could envision moving from a square (four bump wheel), to a three bump,  two bump, one bump, and no bump wheel. Would a one bump wheel be a perfect circle (is a one bump wheel possible?), is a no-bump wheel a perfect circle??





There are many ways in which we can exercise our visual imaginations. We can step back from a mindset and try to visualize the focus of our imaginations in a wholly new way. What, for example, might land vehicles look like if we didn’t use wheels? Or how can we visualize an atom in a way that helps us better understand it’s cloud like nature, as much as it’s quantum states? What if we, for a moment, put away the idea that a plane has to look something like a plane – can it look completely different? There are also many visual illusions that we can look at that might help us experience something of the way our visual imaginations work. One such is that of Salvador Dali ( a face or people standing?).









In some small ways the designs in my Images and Altair design books can also help us exercise our visual imaginations. What can you see in a particular design? Now try and see something else? And again, and again. (From Hidden Images, “Dinosaurs,” published by Running Press.