Am now on page 214 after an edit. Have added a really simple convergence proof for the Fibonacci ratio converging on Phi – so simple that it could even be Pythagorean. Also added an exact and not a fudged analysis of DaVinci’s Vitruvian man with no relation to Phi, just Vitruvius. This is all in the Perspective chapter – an odd place you might think – but it follows nicely after Egyptian grids on the way to perspective drawings, Futurism, Cubism, etc.
This is the new working title after “Universe of Geometries.” Now on page 200 and have continued to add new material to Neolithic, Greek & Pythagorean, Celtic, and Islamic sections. Have more or less completed the Shape Changing Polyhedra section but still have more to add to the Dynamic Close-Packing Sphere section. The latest section is Perspective but have been using it to not only track the development of grids, from Egyptian through the Renaissance, and the present, but also to address the concept of harmony in architectural design using complete sphere packing geometries; where each detail is determined by sphere position. Same would apply to some of the other 3D systems. Might add some irregular 3D curved structures too but need to establish a 3D integrated basis rather than just have casual curves. Still aim to complete the first draft by April 30th although, if I included all my work the book could have 1,000 pages. Over 1,t000 original illustrations so far plus photos.
Am now 110 pages into my new book, A Universe Of Geometries,” with about 200 pages to go. I aim to complete by the end April. The book is content rich, packed with info, and intended as a course, and source, book for architectural and design students, musicians, artists, structural engineers, mathematicians, scientists and those interested in enhancing perception. Am hoping to give guest lectures and run workshops and exhibitions using the book as the foundation. There’s nothing as rich or thorough out there. Final count on the illustrations will be over 1,000 – have drawn them all – plus some photos. Content includes a lot of my geometrical work accumulated over the last 40 years – scary thought that!
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 generated specific proportions that would focus sounds at various parts of the main studio – to enhance perception. So that, with just ambient sounds, you would be able to walk around the room and feel pressure points, almost likes orbs suspended in the air; where natural sounds would resonate. The building is still standing and has the unique property described. In the original design I had also specified blinds of different colors, so that the color properties of the room could be changed, but this was never done.
Whilst staying at Nancy’s house I was surprised to find that she still had one of my old machines built to explore geometries in the late sixties and early seventies. After just a bit of maintenance the machine still works! The machine generates the sort of design structure that I now use Adobe After Effects to generate – in a fraction of the time.
The “Geomorph - When Mathematics Meets Art Everything Changes,” at the Leonardo in SLC was very successful with lots of visitors. The exhibition was organized in four sections: 1. Geometry Through Time; 2. Dynamic Sphere Geometry; 3. Shape Changers; and 4. Images and Altair Designs.
The workshops also turned out to be a great success with visitors decorating panels for a large “shape-changer;” making small shape-changers; constructing Archimedean and Platonic solids; and finding images and designs in the interactive program of geometrical lattices generated by the Dynamic Sphere Geometry.
I met with architectural, math, and science students as well as many people interested in the geometry through time section. Am hoping that the exhibition will serve as a starting point to trigger interest in my geometrical research and systems.
Am now looking to expand the whole concept of the exhibition with LCD projections, bigger LCD screens, more space to exhibit, more working models, much larger, walk-in, shape-changing, and sphere generated architectural forms, – so that the properties of these unique structures can be explored – sound and light reflecting properties, architectural applications – but need sponsors. Also, I’d like to show more in the way of architectural forms and nano-structures – as well as show more of my work relating to geometrical systems of the past.
Some photos of the exhibition are as follows:
For those that visited Geomorph – Thanks for coming!
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!
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 “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.”
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…
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, I have used the geometry to generate about one hundred arrangements – but the geometry should be capable of generating many thousands and, possibly, millions, of new and unique close packing sphere arrangements. For those interested, the whole idea started in 2D by just generating close packing circle arrangements. Circle packing was used by many cultures in the past to generate design and architectural forms: Celtic, Chinese, Islamic, Gothic…
Close-packing spheres are space efficient and have a high structural integrity, but, generally, architects, designers, and those modeling anything from molecules to new types of nano-structure, only have access to a very small range of close-packing sphere arrangements, and to their corresponding lattices. The Dynamic Sphere Geometry takes the lid off this limitation and whole families and series of new space-filling structures can be generated – each with a unique property. For those that don’t know anything about close packing spheres, imagine the centers of the spheres connected to make structural lattices, or even imagine domed buildings, or spherical structures in space, orbiting the Earth, or as nano-particle materials…
In just a few words I cannot communicate the whole geometry, or where it really gets interesting, when new close-packing, and three-dimensional, sphere clusters transform, move in space, and interconnect. Anyway, here’s an introduction by way of an animation and two sets of line drawings, that will hopefully show, just a bit, of how the geometry works. The animation clip shows four transformations (Cells One, Two, Three, and Four), extracted from a series of seven, on a two-dimensional plane.
Press this link for the animation clip: Spheres1b
Fig 1: INTRODUCTION TO DYNAMIC SPHERE GEOMETRY
Cells of close packing spheres are shown where the close-packing spheres of “Cell One” transform into the close-packing spheres of “Cell Two.” These are the first two packings shown in the animation clip. Various rules are imposed on the Dynamic Sphere Geometry that govern how sequences are generated. Spheres repeat infinitely over a two dimensional plane or on multiple planes in three-dimensional space. The spheres of “Cell Two” are in whole number relationhips and two adjacent sides and the diagonal of the rectangle of “Cell Two” are 3, 4, 5 triangles.
Fig 2: SHOWS HOW “CELL TWO” TRANSFORMS INTO A 3D CLUSTER THAT REPEATS INFINITELY IN 3D SPACE. The spheres of “Cell Two,” are in whole number relationships.