My latest book, “3D Thinking,” to be published by Thames and Hudson Spring 2018 forms the basis for a new exhibition, lecture and workshop series, see below.
1: The Golden Ratio – A Divine Proportion? The Golden Ratio – a ratio that, in popular culture, appears in the Great Pyramid of Giza, the Parthenon, and the paintings of DaVinci. It is a fact that the Golden Ratio lies at the end of a convergence with the Fibonacci Sequence (1, 1, 2, 3, 5, 8, 13, 21,..) – a sequence that appears to follow natural growth patterns in plants and many natural events – so does that make it a divine proportion? Does it qualify? Are popular ideas about the Golden Ratio correct or not? How does the Golden Ratio compare with the divine proportions of Vitruvius and the Pythagoreans, as well with the divine proportions of other cultures? This presentation will raise the subject of what might be considered as a ‘divine proportion,’ and will also reveal for the first time properties of the Golden Ratio that are unknown and that might give it an unexpected advantage. 1st Presented at the Leonardo Museum March 2016.
2: The Future of Architecture. “Looking to the future of Architecture we have many amazing opportunities with the possibility of, ‘drive-up,’ massive 3D printers that can create fused silicon, metal, and plastic structures; new types of concrete, bonded wood, new nano- and micro- structured materials; even bio-forming processes of different types. We stand at a threshold where almost any conceivable shape and structure is possible – incredibly cantilevered, cellular, floating, shape-changing, etc. We also have the mathematics and software to generate a huge variety of internal and external forms – with creative transitional spaces. Question is, ‘what will drive new architectural forms?’ Will it be extreme economic constraints, more rectangular boxes – with some high profile sculptural exceptions. Or will it be new types of structure designed to enhance the human experience? Historically there has been some precedence for the latter but not much of an empirical basis and there appears to be very little research into the human impact of, ‘architectural space,’ other than studies in color, light, and ergonomics. So what should the empirical basis be?” Presented at the Leonardo Museum April 2016.
3: Shape Changing Polyhedra. This is an introduction to shape-changing polyhedra – defined as three-dimensional polygonal structures that change shape and size and that create a completely new type of application opportunity. A number of shape-changing geometries are already under development with prototypes that have the potential to make shape-changing buildings, aircraft wings that can flex, two dimensional surfaces that can transform into three dimensional robots, and much more. The geometry presented here is the geometry of shape-changing polyhedra composed of flexibly connected polygons – connected in 2D and 3D arrangements where the 2D will transform into 3D and where 3D will transform from one form into another. Presented at the Bridges Conference August 2016, University of Jyvaskyla, Finland.
4: Arabian Geometries. This is an introduction to concepts that underlie the six main geometric design systems developed during the Abbasid caliphate of the Muslim Empire. The approach aims to reveal design methodologies that can stand independent of the ‘Islamic’ shell that surrounds them – and hopefully will serve as clearer start-points for new concept development. 8th to 13th century CE Arabian geometric design systems evolved due to needs – physical, cultural and philosophical. Designs were often rendered in symbolic form where, for example, the use of rosettes was common, however the methodologies underlying the designs stand independent of any symbolism and can be used creatively where core design lattices can be generated and then serve to generate derivative designs and applications. Methodologies of the early Islamic period can be described as: (i) Grids; (ii) Tessellating Polygon Subdivisions; (iii) Rays; (iv) Close-Packed Circles; (v) Nesting Polygons; (vi) Modular. Layered over the design methodologies are specific color pallets and also function and sequencing. Function can include acoustical, transitional, psychological impact, and even messaging (through the ABJAD number-letter system). ‘Sequencing,’ is the order within which the designs are experienced.
5: The Pythagoreans. This is a presentation of a number of Pythagorean geometrical concepts with speculations about how they might have been further developed at the time of the ancient Greeks even though not much in the way of evidence survives. The intention of the presentation is to show that ancient ideas can still inspire new concepts for present day and future applications. Featured in the presentation will be a review of (i) The Pythagorean theorem and how Euclid’s proof and comments compare with other proofs and concepts particularly a “self-evident” ancient Chinese proof; how the Pythagorean theorem can be interpreted in terms of dynamic circles and how circle configurations cross-over to a unique and unknown close-packing, ‘Golden Rectangle,’ sphere arrangement and its generated tessellating lattices; how Euclid’s ‘mean and extreme’ ratio (‘golden ratio’) also cross-links with the ‘Golden’ close-packing sphere arrangement and the Fibonacci sequence with something of a realistic perspective. (ii) How Platonic and Archimedean solids can be extended and combined and how they might have been used to investigate the emotional / physiological / psychological impacts on men and women of 3D spatial structures (refencing the works of Vitruvius) with some links to the pentatonic scale. (iii) How Pythagorean tessellations combined with Euclidian statements anticipate the non-periodic tilings of Roger Penrose and Islamic periodic tiling designs.
6: A Dynamic Close-Packing Sphere Geometry. This is an introduction to a new and dynamic close-packing sphere geometry. Circles and spheres are part of our everyday vocabulary but their combinations in two and three-dimensional space can still surprise us – with lattices of great potential for nanometer, micro, and macro structural forms: in nanotechnology, in optimum packing configurations, in architecture, and in much more. The dynamic geometry introduced here allows circles and spheres to change size and position moving from one close-packing arrangement to another (of same or different sized circles or spheres). The algorithms of the geometry open up 2D and 3D structural space in a way not envisioned before. Close-packing arrangements are generated in sequences – with a great number of ratios and symmetries. The geometry has so far generated whole number, golden rectangle, Platonic, Archimedean, and hundreds of other close-packing arrangements. An interesting characteristic of many of the packings generated is that they will ‘layer-cake,’ meaning that when planes correspond between one close-packing arrangement and another then packing arrangements can be combined. Another characteristic of some close-packings is that of motion where the removal of selected spheres, within a packing, will allow close-packing cluster arrangements to rotate. There are various sets of algorithms that can be applied where some, for example, maintain symmetrical and circle or sphere boundaries, some maintain enclosing sphere sizes, some can allow sphere penetration, and some add conditions of ‘rates-of-change.’ This is an introduction to the geometry.
7: Labyrinths – Mysteries and Methods. This is an introduction to the mysteries and methods of labyrinths from their Neolithic past and then across time and space through to the present and possible future. The first labyrinths appear as petroglyphs and then more famously in Crete, in ancient Rome, and then as far afield as on islands off the north coast of Russia, the Americas, Scandanavia, and India – and in cathedrals of the 12th and 13th centuries CE. The ‘why,’ and ‘how,’ are the key parts of this presentation. The ‘why,’ ranges from mere doodling to ancient Earth concepts of energy and passages to the underworld, to solar and lunar calendars, to paths for dancing, meditation, and chanting. The ‘how,’ is presented by way of various methods for generating labyrinths from the ancient to the more modern: (i) Key; (ii) Grid; (iii) Modular (iv) Switch.
8: Geometries of Pre-Columbian America. This is an introduction to the geometries of the Pre-Columbian Americas with an overview of astronomical and architectural practices and concepts. The first humans appear to have arrived sometime after the last ice-age, possibly 14,000 BCE, expanding and diversifying into tribal groups numbering in their thousands with an equivalent number of languages – possibly evolving from a core group of about fifty. Many civilizations and cultures arose throughout the Americas including the Pueblo, Adena, Hopewell, and Mississippian cultures in the North, the Aztec and Mayan civilizations in middle America, and the Nazca, Muisca, and Inca cultures and civilization in the South. Given the diversification a great number of architectural, geometric, astronomical, and numeric methods and practices developed and many are presented with the view of promoting discussions.
9: Indus Valley Geometries
“3D Thinking” – “An exploration of visual logic through time and space.”
The foundations, forms and patterns in architecture, design and decorative arts throughout the world have been deeply influenced by the geometries of past cultures. From the first path-like doodles on cave walls through to the higher abstractions developed to make accurate measurements and predictions, the three-dimensional forms we design and build are dependent upon available materials, human needs and the limitations of our imaginations.
The product of decades of teaching and research, this unique and inspirational design resource presents a history of the intimate relationships between geometry, mathematics 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. Explaining key principles that can be applied across all design disciplines, Roger Burrows 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 and professionals.
- 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 hundreds of specially created diagrams and artworks.