Reviewed: The New Mathematics of Architecture

The book is a collection of stories about projects, each of which includes a textural and a visual description of the project from both mathematical and architectural points of view.  The first thing I noticed about the book is how beautiful the projects are.  I don’t know how much of that is related to them being mathematical projects, because they are also generally beautiful in their own right.

After reading this book, the overwhelming impression is that mathematics is a vital tool for architecture and that architecture changes as maths does.  The most obvious example of this relationship is seen in the 2D and 3D CAD programs which underlie modern architectural practice and which operate almost entirely on mathematical principles.

Mark and Jane Burry provide a compelling case, backed up with well researched and informative examples, that the connection is, or rather can be, much deeper.  Many of the individual projects described are lacking in some of the detail which some readers may desire regarding the mathematical processes used or the realised architectural details, but they do provide an interesting taster to whet the appetite.

The body of the book comprises six sections, each describing a different area of maths and how it is applied to architecture.  These sections are tied together by insightful opening blurbs which introduce the topics which are exemplified by projects.  These blurbs are a chance for the reader to learn and reflect. They set the context in which Jane and Mark see the projects. In spite of the astonishing beauty of the projects described, these blurbs are in my view, the best parts of the book.  The projects then provide further evidence and depth to the existing ideas.

The first section, Mathematical Surfaces and Seriality, may well be the strongest section architecturally, but is also perhaps the least unified mathematically. In addition to introducing sequences and series, it also covers in some detail the idea of mathematical surfaces.  Maths deals with abstractly beautiful things which may be inspired by, but need not be directly related to, physical objects.  A “surface” in mathematical terms is a conceptual object which has no depth.  These can be studied in a variety of ways, some of which are detailed in a later chapter.  Here, we are introduced to minimal surfaces, ruled surfaces, and Antoni Gaudi.  Minimal surfaces are surfaces where every point is a special type of “saddle point,” curving up in one direction and down in the other.  If you take a wire frame and dip it in bubble mixture, the resulting surface will be minimal.  Ruled surfaces are basically those which can be built out of pieces of paper by bending without scrunching or cutting, used to make projects like Gehry’s Disney Concert Hall buildable.  And Antoni Gaudi was an architectural genius.

The first project described is the Australian Wildlife Health Centre (Minifie van Schaik Architects) at the Healesville Sanctuary which was based on a minimal surface found in 1984, with a computer choosing the parameters to ensure that the building could handle the programmatic requirements.

Maths is a whole lot bigger and more dynamic than most people would suspect.  Many people would be surprised to learn that a new surface was recently discovered.

In maths, there are some questions we know the answer to, some we know we can never know, and lots  that we’re still trying to figure out.  Classifying all of the minimal surfaces which fit into three dimensions is a recently solved problem, as shown in a 2008 paper by Charles Frohman and William Meeks.

One of the other stand-out projects in this section is the Segrada Familia, a building with an incredible back story which started construction in 1882 and is still being built.  Although he wasn’t the first of the Segrada Familia’s architects, Antoni Gaudi was instrumental in its design.  Mark and Jane explain and show examples of how Gaudi uses mathematics as a tool to design incredible objects.  By adding and subtracting simple three dimensional shapes in different locations and with different angles, he is able to create structures where we naturally read the missing objects and which have incredible complexity.  The other key design feature which is used throughout is the idea of using near copies of each other.  Making objects which can easily be read as belonging to the same family, but which are different enough to have individual identities, leads to the completed building providing the occupant with a more natural experience than when objects are more regimented.

As with Gehry’s ruled surfaces, Gaudi is using neat mathematical ideas to describe objects which become part of our environment.  This chapter shows us something which is perhaps even deeper than the particular pieces of mathematics which are being used.  It shows us that those who are comfortable with mathematical ways of thinking, understanding, and describing the world can use this to achieve amazing results in the built environment.

Section 2 introduces the closely related topics of chaos, complexity, and emergence.  Contrary to popular opinion, chaos theory is a very specific area of maths, relating to the study of systems which exhibit “extreme sensitivity to initial conditions” and so a very small change in the state of the system can result in a very different final outcome.  The most common example of a chaotic system is weather and the well known “butterfly effect”.  Complexity and emergence refer to systems made up of simple parts which work together to make complex and beautiful results.

The first project described in this section is Melbourne’s Federation Square.  The characteristic triangular pattern is what’s known as a fractal, a shape or pattern which looks the same no matter how far you zoom in.  Again, the mathematical idea is clean and pure.  If you want to actually create something, you can’t make the zoom go down to infinitely much detail and so there is a smallest triangle on Federation square.  The pattern on Fed Square is made up of small triangle which go together to make slight larger versions of the same triangles. This process can be repeated until you have all of Fed Square covered.  Some of the best known examples of fractal like structures in nature are ferns and the line between the ocean and land.

The other stand out project in this section is the Louvre Abu Dhabi by Ateliers Jean Nouvel which is currently under construction. The museum sits under a 180m diameter dome with many simple lattice patterns letting light through.  These lattices of different sizes are rotated and layered leaving a spectacular random seeming series of areas where light can enter the building from above.

The third section of the book, packing and tiling is perhaps one of the more obvious areas where mathematics and architecture can combine and has some links to the complexity of the second section.  Here the maths deals with fitting objects in to space as closely as possible.  Of course, for the mathematicians, it doesn’t have to be three dimensional space.  A recent talk at The University of Melbourne discussed packing oranges in 24 dimensions.

The Water Cube from the Beijing Olympics by PTW Architects looks like bubbles stuck inside a box.  Not only is it a striking and memorable structure, but it is also environmentally and structurally sound due to its lightweight construction.  The outer surface of the Cube is a beautifully complex combination of different sizes and shapes which comes from a relatively simple idea.  It uses a 3D packing of similar shapes for easier construction, and then rotates it away from our usual three axes.  Taking vertical slices through this rotated structure is what gives the seemingly random pattern.

The Spiral Extension is an unbuilt competition winner which promised to make its mark on the street.  I found myself enthralled by the description of the aperiodic tiling (a tiling which can fill up space, but which can’t be slid on top of itself symmetrically) which forms the façade of this building.  It has intricate and interesting mathematical properties.  This is one of many examples where the mathematician and the architect were reading the book differently.  I haven’t yet mentioned the spiral.  The structure of the extension spirals out over the pedestrian walkway like a Transformer halfway between car and robot.  In this building, there are two separately interesting ideas which suffer somewhat from the risk of being dominated by the other.

The Victorian College of the Arts Centre For Ideas by Minifie Nixon expresses a mathematical concept which is both pretty and useful, called Voronoi diagrams.  Each Voronoi diagram is defined by a set of points.  Imagine a piece of paper with a number of dots drawn on it in purple pen by a nearby three year old.  For each of the purple dots, you can find the set of white points which are closer to it than to any other purple dot which we call a cell.  If you draw the boundaries of all of the cells, what you have is a Voronoi diagram.  These shapes will all be defined by straight lines, assuming you had only finitely many purple dots.

an example Voronoi diagram

The Centre for Ideas accentuates the Voronoi diagram with large metallic spheres marking our purple dots, sloping the individual cells inward toward them, and building everything out of reflective metal.  To me, the result is not an elegant expression of this idea.  It feels like there are far too few chosen points, leaving cells which are on a scale well above human scale.  To me the façade reads as a large, shaped, reflective surface rather than as a Voronoi diagram.  A large reflective surface feels like such a big statement that it should be the whole story, whereas the Centre for Ideas has many other elements which seem to fight for the focus.

Some of what is fun about Voronoi diagrams is the way the diagram responds to changing the locations and number of points.  In my opinion, the variety of these diagrams needs far more cells scattered throughout the building in order to see it in more detail.

A final noteworthy project from this chapter is the Grand Egyptian Museum which reflects the nearby pyramids with giant triangles.  In particular, the triangles are large and imposing Sierpinski triangles, a fractal which theoretically mirrors itself down to the infinitesimally small, playing with size within view of one of the world’s most impressive structures.

The fourth section, Optimization (which should be spelt Optimisation) was the clincher for me to buy this book.  My PhD is in the field of optimisation.  I work on problems which appear in business or industry where you choose a set of decisions to ensure the best overall value of the outcome.

One of the many ways to classify optimisation methods is as either exact or heuristic.  Exact methods are guaranteed to find the best solution in a finite amount of time (but possibly quite long – occasionally measured in lifetimes of the universe).  Heuristic methods do not enjoy such guarantees.  They are often inspired by natural processes such as evolution.

In this section, it appears that most of the methods used are fascinating heuristics which aim to provide structural support with minimal materials and weight.  What differentiates these structures is the addition of an extra goal.  The optimisation is usually asked to find something which matches closely to a shape or a usage defined by the architect.

Qatar Education City Convention Centre

A great example of a beautiful result from optimization is the tree-like structural supports of the Qatar Education City Convention Centre.  To design structural supports, they start with a conceptual support which fills all of the available space and carve out unnecessary portions until everything that is left is essential.  To choose which piece to carve out in each time period, the structure is simulated and an area with the minimum force through it is removed.

The fifth section of the book is about a mathematical area known as topology.  Maths is about abstractions, and topology is one of the more abstract areas of maths.  In topology, we consider shapes and surfaces but we abstract away notions such as length, dimensions, size, and edges.  You can pinch, stretch, and squeeze shapes and surfaces all you like without changing them topologically.

As beautifully described by Jane and Mark, there are times when this degree of abstraction allows you to simplify problems without losing their essence.  This simplification can allow you to see connections to other problems and to find solutions.

Probably the most interesting topological objects to architects are 2D manifolds.  These are closed two dimensional surfaces, such as the surface of a beach ball or the surface of a donut.  Not all 2D manifolds fit neatly into our usual three dimensional space.  The classic example is the klein bottle, a shape with no inside or outside which has to cross through itself when embedded into three dimensions.

I must admit to having found this section somewhat disappointing.  Topology is about abstract shapes and so any physical construction of them must limit what they could have been.  That said, the idea of these shapes seems to be inspirational to architects.  I remember sitting next to the Red and Black Architect in a lecture about topology and his reaction to seeing a Klein bottle was to start sketching a Klein bottle house.

In the book, there are houses based on both the Klein bottle and the well known Mobius strip.  The discussion of these projects is fantastic.  Mobius strip house talks about eternal flow around a loop between day and night.  Klein bottle house is about playing with inside and outside.  I, personally, was unable to read these ideals from the images of the houses provided.  The addition of hard edges to shapes which are traditionally visualised as curved does not violate their topological identity but makes them harder to read.

The stand out project in this section is another competition winner, Mobius Bridge by Hakes Associates.  This bridge is a sleek and sexy pedestrian bridge which lives separately to the banks around it, speaking its own language without interfering with the sights and views around it.

Mobius Bridge by Hakes Associates

The final section is titled Datascapes and Multi-Dimensionality.  Data, and in particular “Big Data” is a growing field at the intersection of maths, statistics, and computer science.  Visualisation of ever growing streams of data is an important research field in its own right.

Visualisation can serve one of two main purposes.  It is usually designed either to better understand and express data or to show something cool, such as singing the digits of pi.  Integration of data visualisation with architecture runs the risk of being all about the hype of showing something new and shiny.  Now don’t get me wrong, I love a cool maths thing as much as the next person (probably more, given that the next person probably isn’t really in to maths), but there are ideas presented in the book which provide a much deeper connection between maths and architecture than the idea of showing something cool on the façade of a building.

The projects in this section were largely a pleasant surprise.  The Hotel Prestige Forest by Cloud 9 Architecture is covered in “leaves” which are each a photovoltaic cell absorbing sunlight during the day and releasing the gathered energy as a colour reflecting the average temperature on that leaf throughout the day.  The leaves don’t need to communicate with each other, the colour on the building will inherently vary across the building and across the year.  Here in the last section is the first time I felt like I strongly disagreed with Jane and Mark where they seemed to suggest that the leaves should interact.  In my opinion, the story is better because the leaves can are acting independently and each is, in a way, describing its own experience.

The other project type which I found fascinating from this section were the building forms, or even whole rooms, which physically change their structure in response to the people in and around them.

Perhaps the prettiest and definitely the most surprising interactive form is the Digital Water Pavilion by Carlo Ratti Associati.  The pavilion is fronted by a falling wall of water jets.  By turning each jet on and off, the water and the gaps between it can be used like a screen to show falling images.

[youtube=http://www.youtube.com/watch?v=SBDaYfLORPs]

When I first saw this book, I agreed to write a book report for the blog.  I have ended up writing a lot more than I intended.  Overall, this is a fascinating and beautiful book which showcases the authors’ in depth knowledge of both fields.  Architecture has always used maths to describe the structures it wants built.  As maths and the computers we now rely on develop and we can express more, the types of architecture which can be built expands.  The key take away from this book, however, is that we can use a deep knowledge of mathematical ideas and ways of thinking to go beyond maths as a tool for expression, to maths for design, maths generating designs, maths as inspiration, and to using the built environment to express deep mathematical ideas in surprising and intricate ways.

About the Author

Dr Olivia Smith is a Mathematician specializing in Operations Research. As well as being an architecture enthusiast, she is also a lecturer at the National Mathematics Summer School.