# Math and art relationship

### ART+MATH=X | relax-sakura.info

Compasses, rulers, grids, mechanical devices, keyboard and mouse are physical tools for the creation of art, but without the power of mathematical relationships. Annual conference shines a spotlight on mathematical art and artistic is the study of patterns, structures, relationships," says George Hart. In her new book Mathematics and Art, historian Lyn Gamwell explores how artists have for thousands of years used mathematical concepts.

We can perceive it almost subconsciously — and it has been argued that it plays a vital role in our perception of beauty — yet it opens the door to a wealth of mathematical structure. A square, for example, has 8 symmetries: Each of these transformations is called a symmetry, because after you've done it, the square appears to be exactly as it was before.

If you put all these 8 symmetries together, you get a self-contained system: Such a self-contained system of symmetries is called a group, and symmetry groups are the gateway to abstract algebra.

A simple visual consideration lands you in the thick of some quite advanced mathematics! The Von Koch Snowflake The Von Koch snowflake is a fractal which is constructed from an equilateral triangle as follows: If you keep going indefinitely, you end up with the fractal shown on the right.

The outline of the snowflake is incredibly crinkly. In fact, it does not contain any straight line pieces at all: Mathematicians measure the crinkliness of a fractal by the fractal dimension, a generalisation of our ordinary notion of dimension.

The outline of our snowflake is too crinkly to be one-dimensional. On the other hand, it clearly is not two-dimensional either, since it contains no area. So, clearly, the visual arts can provide a very good entry to maths for non-mathematicians, but how would mathematicians or maths students benefit from a visual arts course?

I remember when I did my PhD, the professor would sometimes sketch something on the board to aid understanding, but those sketches usually got discarded pretty quickly.

• Mathematics and art
• Why the history of maths is also the history of art
• ART+MATH=X

I think that, especially with the help of computers, people could learn how to "see" things they previously found hard to imagine. Colours, for example, can be used to convey a lot of information. Textbooks often only contain black-and-white pictures, if any. I think that someone skilled in the use of colours could create diagrams that represent the ideas much better".

This is indeed something that has been done in some areas of mathematics. The famous Mandelbrot set, for example, is so colourful not only because it is prettier that way.

The colours also have an exact mathematical meaning. A detail of the Mandelbrot set.

## Math in Art

Carla thinks that it is time to rethink the way we teach maths. I think there are many maths teachers who are not too aware of the ideas behind the maths and simply teach equations and formulae. So people usually don't see maths as a creative subject. Also, mathematics is hard, and if people find it difficult and not creative — well, of course they won't like it! We should think very hard about education and try to focus on ideas, rather than technicalities. Again, computers and visual representation could be pivotal in this.

That's why I set up the Special Year; I really wanted to make people aware of what doing maths is really like, and that it has creative aspects. You can admire some of the visual exhibits on the digital gallery. In some of the works the connection to maths is clear: Sri Yantra, Erin Russek, fabric. Other exhibits, however, are not so clearly connected to maths. Sarawut Chutiwongpeti's video installation Untitled Wishes, Lies and Dreams explores the mechanisms underlying perception and the world of dreams and the unconscious.

Does the unconscious play a part in maths, too? It's the place from where we all bring our ideas. I think there is a connection between dreams and maths. Dreams have their own logic, that doesn't immediately make sense. In mathematics, too, we use many different types of logic, sometimes counterintuitive ones. Does the unconscious play a part in maths?

### Bridging the Gap Between Math and Art [Slide Show] - Scientific American

It's about taking body pieces and attaching them to parts of the body where they don't belong. This is something that could happen in a dream, but it's also very systematic: Both maths and dreams allow us to break the boundaries of reality. Maybe it's all about expressing the infinite world we have inside of us, and which in our real life can only come out in finitely many ways.

The exhibitions of visual art form only one aspect of the Special Year. There are various plays and concerts, each stressing that maths — and mathematicians — do not exist in an isolated world of books and blackboards. It describes the life of Alan Turing, whose mathematical genius helped to break the Germans' Enigma code during the Second World War, but whose homosexuality meant that he was also breaking the moral code of the time.

He was tried and punished for what the authorities considered his crime, and ultimately committed suicide. According to the philosopher and mathematician XenocratesPolykleitos is ranked as one of the most important sculptors of Classical antiquity for his work on the Doryphorus and the statue of Hera in the Heraion of Argos.

In the Canon of Polykleitos, a treatise he wrote designed to document the "perfect" anatomical proportions of the male nude, Polykleitos gives us a mathematical approach towards sculpturing the human body. This geometric series of measurements progresses until Polykleitos has formed the arm, chest, body, and so on.

While none of Polykleitos's original works survive, Roman copies demonstrate his ideal of physical perfection and mathematical precision. Some scholars argue that Pythagorean thought influenced the Canon of Polykleitos.

In the Middle Ages, some artists used reverse perspective for special emphasis. The Muslim mathematician Alhazen Ibn al-Haytham described a theory of optics in his Book of Optics inbut never applied it to art.

Two major motives drove artists in the late Middle Ages and the Renaissance towards mathematics. In the hands of an artist, mathematically-produced art is only a beginning, a skeleton or a template to which the artist brings imagination, training, and a personal vision that can transform the mathematically perfect to an image or form that is truly inspired.

Wallpaper patterns and tessellations can be pleasing from a decorative point of view; few would be viewed as art. Escher did not view his tessellations as art, but as fragments to be an integral part of his complex prints. Makoto Nakamura's art also employs this technique. Pure mathematical form, often with high symmetry, is the inspiration for several sculptors who create lyrical, breathtaking works.

With practiced eye and hand, relying on their experience with wood, stone, bronze, and other tactile materials, the artists deviate, exaggerate, subtract, overlay, surround, or otherwise change the form into something new, often dazzlingly beautiful.

With the advent of digital tools to create sculpture, the possibilities of experimentation without destruction of material or of producing otherwise impossible forms infinitely extends the sculptor's abilities. Yet many mathematical constraints cannot be rejected; artists ignorant of these constraints may labor to realize an idea only to find that its realization is, indeed, impossible. Other theorems govern the topology of knots and surfaces, aspects of symmetry and periodicity on surfaces and in space, facts of ratio, proportion, and similarity, the necessity for convergence of parallel lines to a point, and so on.

Rather than confining art or requiring art to conform to a narrow set of rules, an understanding of essential mathematical constraints frees artists to use their full intuition and creativity within the constraints, even to push the boundaries of those constraints.

Constraints need not be negative -- they can show the often limitless realm of the possible. Voluntary mathematical constraints can serve to guide artistic creation.

Proportion has always been fundamental in the aesthetic of art, guiding composition, design, and form. Mathematically, this translates into the observance of ratios. Whether these be canons of human proportion, architectural design, or even symbols and letter fonts, ratios connect parts of a design to the whole, and to each other.

Repeated ratios imply self-similarity, hardly a new topic despite its recent mathematical attention. One of the earliest recorded notices of it is in Euclid's Prop. Other ratios and special geometric constructions root rectangles, reciprocal rectangles, and grids of similar figures also guide composition and design.

This goes with the territory. In many instances, artists will struggle to answer the questions on their own in order to reach the answer in a way that makes sense to them. Escher did this in seeking to answer the question "How can I create a shape that will tile the plane in such a way that every tile is surrounded in the same way? The intricate textile patterns of designer Jhane Barnes result from close collaboration with mathematician Bill Jones and computer software designer Dana Cartwright of Designer Software.

Folk art from other times and other cultures is a rich source for mathematical questions. Celtic knots and art from African cultures are two examples. Two of these mathematical questions seek to understand the relationships between local and global symmetry. A most mathematical artist I want to end this essay with a bit more about the work of the Dutch graphic artist M. Escherwho is perhaps the most astonishing recent example of an artist whose work contains a multitude of connections between mathematics and art.

Yet he did not reject mathematics, but instead figured out in his own way, using various mostly pictorial sources, the mathematics that he needed in order to realize his ideas and visions.

Escher celebrated mathematical forms: