Kas Oosterhuis speaks about a new kind of building and about the way architecture will be influencing our understanding of what’s beautiful and what is not, towards a new kind of beauty.

Kas was born in 1951 in Amersfoort (Netherlands), he studied architecture at the TU in Delft. Afterwards, he taught as unit master at the AA in London. From there, he worked and lived one year in the former studio of Theo van Doesburg in Paris, together with visual artist Ilona Lénárd. In 1989, he founded Kas Oosterhuis Architekten in Rotterdam (renamed to Oosterhuis Lénárd, or ONL, in 2004). Since 2000, Oosterhuis has been professor of digital design methods at the Delft University of Technology.

Block aims to democratize design by helping citizens brainstorm creative but realistic ideas for their own urban spaces.

More than 100 million people play Minecraft, building virtual worlds out of virtual blocks. What if some of them started to use the same skills to redesign actual cities?

A new video game in development at the University of Southern California’s School of Architecture is designed to make anyone a city planner. Using real city data, the game will start with Los Angeles but has the potential to be tailored to any city in the world.

“Gamescapes research seeks to democratize the design of the built environment through the use of video games,” says Jose Sanchez, assistant professor in architecture at USC, who is working on the project with Satoru Sugihara and Sergio Irigoyen.

In the game, called Block, players build neighborhoods from components like housing, businesses, and parks, continually rearranging the pieces to find the optimum arrangement that citizens would most want to use and that would have the least cost for the city.

The game also measures environmental costs and benefits–like whether a particular arrangement can generate energy or produce food for the neighborhood. Sanchez was particularly inspired by a Chicago vertical farming project called the Plant, which connects a system of businesses so that each can use waste and energy from another.

“The Plant’s great innovation is to look at how the different elements of production feed into each other,” says Sanchez. “If we have hundreds of different elements that you can combine in different ways, people could just start finding interesting combinations–let’s say combining a bakery with something else that would eliminate waste.”

The game is not about details of architecture or aesthetics, but about how various parts relate to each other. “That’s kind of the ultimate goal of the game, to empower people to speculate on what a city really needs,” says Sanchez.

As much as possible, Block includes real data about each part of a system. “If we know, for example, how much a certain solar panel costs and how much energy it produces roughly, we can give a quick estimate,” Sanchez says. “Those are indicators that an audience could use to start playing the game and really coming up with creative solutions.”

The game will be open-source and free to play, and as players come up with various designs, they’ll be able to add to a growing database of designs that a city could actually implement. The researchers will also be able to pull data from the game and look for patterns in citizen ideas that official urban planners might be overlooking in their own work.

The researchers say the game would be open to everyone; a 10-year-old could have ideas as interesting as someone with a degree in planning. “If we present things in an easy way, but they potentially have a lot of complexity if you dedicate time to the game, you could have a very low entry level but really accumulate a large audience,” says Sanchez.

An early prototype of the game was first developed a few months ago for a conference in Hong Kong, where the researchers looked at different factors like vertical growth and rooftop farming. The designers hope to develop it for multiple cities. The focus now is on L.A., and the design is currently one of the finalists in the LA2050 grants challenge, a contest that will award $1 million to 10 projects that help improve the city.

The game will be released in about a year, but a very early prototype version is available for download now.

The work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians who have obsessed about Platonic solids, a class of geometric forms that are highly regular and are commonly found in nature.

Since Plato’s work, two other classes of equilateral convex polyhedra, as the collective of these shapes are called, have been found: Archimedean solids (including truncated icosahedron) and Kepler solids (including rhombic polyhedra). Nearly 400 years after the last class was described, researchers claim that they may have now invented a new, fourth class, which they call Goldberg polyhedra. Also, they believe that their rules show that an infinite number of such classes could exist.

Platonic love for geometry

Equilateral convex polyhedra need to have certain characteristics. First, each of the sides of the polyhedra needs to be of the same length. Second, the shape must be completely solid: that is, it must have a well-defined inside and outside that is separated by the shape itself. Third, any point on a line that connects two points in a shape must never fall outside the shape.

Platonic solids, the first class of such shapes, are well known. They consist of five different shapes: tetrahedron, cube, octahedron, dodecahedron and icosahedron. They have four, six, eight, twelve and twenty faces, respectively.

These highly regular structures are commonly found in nature. For instance, the carbon atoms in a diamond are arranged in a tetrahedral shape. Common salt and fool’s gold (iron sulfide) form cubic crystals, and calcium fluoride forms octahedral crystals.

The new discovery comes from researchers who were inspired by finding such interesting polyhedra in their own work that involved the human eye. Stan Schein at the University of California in Los Angeles was studying the retina of the eye when he became interested in the structure of protein called clathrin. Clathrin is involved in moving resources inside and outside cells, and in that process it forms only a handful number of shapes. These shapes intrigued Schein, who ended up coming up with a mathematical explanation for the phenomenon.

During this work, Schein came across the work of 20th century mathematician Michael Goldberg who described a set of new shapes, which have been named after him, as Goldberg polyhedra. The easiest Goldberg polyhedron to imagine looks like a blown-up football, as the shape is made of many pentagons and hexagons connected to each other in a symmetrical manner (see image to the left).

However, Schein believes that Goldberg’s shapes – or cages, as geometers call them – are not polyhedra. “It may be confusing because Goldberg called them polyhedra, a perfectly sensible name to a graph theorist, but to a geometer, polyhedra require planar faces,” Schein said.

Instead, in a new paper in the Proceedings of the National Academy of Sciences, Schein and his colleague James Gayed have described that a fourth class of convex polyhedra, which given Goldberg’s influence they want to call Goldberg polyhedra, even at the cost of confusing others.

A crude way to describe Schein and Gayed’s work, according to David Craven at the University of Birmingham, “is to take a cube and blow it up like a balloon” – which would make its faces bulge (see image to the right). The point at which the new shapes breaks the third rule – which is, any point on a line that connects two points in that shape falls outside the shape – is what Schein and Gayed care about most.

Craven said, “There are two problems: the bulging of the faces, whether it creates a shape like a saddle, and how you turn those bulging faces into multi-faceted shapes. The first is relatively easy to solve. The second is the main problem. Here one can draw hexagons on the side of the bulge, but these hexagons won’t be flat. The question is whether you can push and pull all these hexagons around to make each and everyone of them flat.”

During the imagined bulging process, even one that involves replacing the bulge with multiple hexagons, as Craven points out, there will be formation of internal angles. These angles formed between lines of the same faces – referred to as dihedral angle discrepancies – means that, according to Schein and Gayed, the shape is no longer a polyhedron. Instead they claimed to have found a way of making those angles zero, which makes all the faces flat, and what is left is a true convex polyhedron (see image below).

Their rules, they claim, can be applied to develop other classes of convex polyhedra. These shapes will be with more and more faces, and in that sense there should be an infinite variety of them.

Playing with shapes

Such mathematical discoveries don’t have immediate applications, but often many are found. For example, dome-shaped buildings are never circular in shape. Instead they are built like half-cut Goldberg polyhedra, consisting of many regular shapes that give more strength to the structure than using round-shaped construction material.

However, there may be some immediate applications. The new rules create polyhedra that have structures similar to viruses or fullerenes, a carbon allotrope. The fact that there has been no “cure” against influenza, or common flu, shows that stopping viruses is hard. But if we are able to describe the structure of a virus accurately, we get a step closer to finding a way of fighting them.

If nothing else, Schein’s work will invoke mathematicians to find other interesting geometric shapes, now that equilateral convex polyhedra may have been done with.

Update: The post was corrected to clarify that it refers to equilateral convex polyhedra.

AUTHOR

Akshat Rathi

Science and Data Editor at The Conversation

INTERVIEWED

David Craven

Birmingham Fellow at University of Birmingham

Stan Schein Professor at University of California, Los Angeles