Thursday, December 23, 2010

Hyperbolic geometry in Roman mosaics?

The Real Alcazar in Cordoba, Spain contains a wonderful collection of mosaics from the Roman period. The patterns used are geometrical in nature and I have already shown some examples of the impossible Thiery-figure in one of the other blog posts. One large mosaic caught my attention because it contained some images I did not really expect to see.

The mosaic displayed against the wall
Close-up of the border of the mosaic

A close-up of the mosaic shows these images as the edge of the mosaic whose edges are semicircles.
Zooming in on one individual image we see clearly that the top of the black region is a semi-circle and the bottom is made up of two smaller semi-circles.

Hyperbolic geometry is one possible non-Euclidean geometry (another is spherical geometry). There are several different ways to model this space. One of those is the so-called "Upper half-space Model". Instead of using the whole plane like we do in Euclidean geometry, we only use the upper half of the plane (as suggested by the name).

In Euclidean geometry we draw images of polygons using straight line segments. These segments are the shortest possible curve between two points. In hyperbolic geometry the sides of polygons are made up of vertical straight lines or semi-circles that meet the bottom of the half-space in a right angle.
In other words, the black object shown above is a hyperbolic triangle!

It takes a little getting used to, but in hyperbolic space the angles are smaller than the ones we see in Euclidean geometry. This triangle has three very small angles and the sum of the angles of a triangle is definitely not 180 degrees in this geometry. In hyperbolic geometry the sum of the angles in a triangle is actually always strictly smaller than 180 degrees.

Wednesday, September 29, 2010

Fractals, Infinity or Both?

The Real Alcazar - or Royal Palace - in Seville, Spain has an area at one of the lower levels called "The Queen's Bath". It would have been a pretty large (and shallow) bathtub to use for one person and the most interesting feature of this room is actually the architecture surrounding the bath. The ribbed faults that support the roof over the bath create an almost hypnotic effect. We see ever more arches contained within the arches as our eye travels to the back of the space.

The Queen's Bath in the Real Alcazar, Seville, Spain
One way to think about this image is to think of it as suggesting infinity. Behind each arch in another arch, and behind that arch is yet another one, and so on and so on. We have to fool ourselves a little bit, because we obviously know very well that the space does not go on indefinitely.

Another way to think about the image is to consider the fact that the photograph contains smaller copies of itself. This is called self-similarity and is one way to create fractal images. In a perfect fractal there would indeed be infinitely many copies of the original image contained in the photograph. In this example we see a partial realization of such a self similar figure.

Below is a photograph of a fairly narrow hallway next to the Queen's Bath. We again see this illusion of infinity in a doorway that contains an image of another doorway, containing yet another image of a doorway, etc. There are at least 5 doorways in this image and one could imaging that there would be infinitely more behind them, thereby creating an infinitely long corridor with infinitely many doorways.

Narrow hallway next to the Queen's Bath
I wanted to complete this (short) gallery of photographs with this image from the Mezquita in Cordoba, Spain. In this photograph we yet again see this illusion of infinity. In this instance there is a suggestion of an infinite number of arches receding into the distance.

Arches in the Mezquita in Cordoba, Spain

Any student of art would interpret this as a case of linear perspective with the vanishing point located close to or at the center of the image of course. 

Friday, September 17, 2010

Escher's pillars in a high school in The Hague, Netherlands

A couple of years ago I spent a couple of weeks in Holland and decided to look for Escher's designs. I knew from my readings that Escher designed three pillars for the central auditorium of the Johanna Westerman School in 1959.  Escher did this project in collaboration with an architect named Bleeker.

The pillars support a balcony in the auditorium of the school. The school and auditorium are still in use and when we arrived they were in the middle of a graduation ceremony. When we returned later that day we had a chance to freely look around and take pictures of the pillars.

The 3 pillars in the Johanna Westerman School 

The tiles were made by "De Porceleyne Fles", a tile and ceramics company from Delft. The three designs were carefully chosen by Escher to be educational.

The front column has rotational symmetry: consider the white lizard and give it a half-turn around the point defined by his nose. The lizard will then align perfectly with the other white lizard. The design consisting of lizards is based on regular division print 104 as shown in Schattschneider's Visions of Symmetry.

The pillar can be tiled with just one type of tile in this case. Each tile consists of one while lizard and the rest of the tile is black.  A half-turn on alternate tiles then creates the pattern we see. The black segments carefully match up to create the image of the black lizards.

The second column is based on regular division print 74 and from the point of view of symmetries is the simplest of the three. The design only has translational symmetries. We can slide one image on top of another, but no other motions will cause the separate images to align themselves.

Here again only one tile is used. In this case the tile does not contain a full image of a bird, but instead is carefully designed to create images of black and while birds when columns of tiles are placed with a vertical shift allowing the heads of the birds to align with their bodies.

The last column is based on print 96 and has glide reflectional symmetries. If you take the image of a black swan and shift it up half a tile length and then flip it over a vertical line, it will align itself with the white swan.

One tile is used to cover the pillar and create a pattern where white swans fly to the right, while black swans fly to the left.

[Thanks to my sister for her company, support and for several of these photographs.]

1. Schattschneider, Doris [1990] (2004). Visions of Symmetry - Notebooks, Periodic Drawings, and Related Work of M. C. Escher, 2nd, Harry N. Abrams. ISBN 0810943085.

Ambiguous Figures - The Thiery Figure

While in Spain this summer I came across an ambiguous figure now called a Thiery figure. It is clear from the presence of this shape in mosaics from the Roman period on display in Cordoba, the floor of the Giralda in Seville and the back of a bench in the Real Alcazar in Seville that this ambiguous figure has been around for a long time.

Thiery figure in a mosaic from  the Roman period. (Cordoba, Spain)
The ambiguity of the figure allows us to view the square in the center as a face of a cube to its left as well as the face of a cube to the right. Consider the square in the center, and focus on the vertex at the bottom. This vertex is simultaneously the bottom left corner of the cube to the left and the bottom right corner of the cube on the right.

Bench in the Real Alcazar in Seville, Spain.
 The surface of this bench in the Real Alcazar in Seville, Spain is rather weathered, but the tiling on the top part of the bench is another ambiguous shape. In this case, due to the scale, it can make the pattern appear as a collection of elongated blocks all pointing to the upper left, or blocks pointing to your lower right.

This final example comes from the floor of one of the chapels in the Giralda in Seville Spain.

It is again the Thiery figure. Located on the floor it leaves one with the impression one is walking on a collection of cubes. But in this case are the tops of the cubes made of the white faces, or the red ones?

The name - Thiery figure - is said to have been given in 1895 when it was named after the psychologist A. Thiery. The examples here show that the figure has been around a lot longer than that.