2+2 = 1? Patterns in Modular arithmetic

November 20, 2011

When someone is talking about the absolute truth of mathematics and declares that once you have defined 2 and +, then 2+2 must equal 4, there is a slightly glib response:

but 2+2 = 1…Mod 3

Despite this surprise, we actually all use modular arithmetic regularly, quite literally on a daily basis. When we consider six hours after 8am, the answer is not 14, but 2pm. Well you could argue for using a 24 hour clock, but no one would claim that 3am on a Tuesday morning is really 27:00 on Monday (well apparently some do, thanks to kuromagi on reddit for ref) In these cases we are not counting as we usually do, but counting on a circle mod 12 or 24. It is not hard to see that we could do this with other numbers. if we do decide that 2+1 is 0, and not 3 we are now working mod 3. In this case 2+2 is 1, as is 2*2. We can put together a small table:

+ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1

Showing what happens when the values for the column and row are added together. We can make the same table for multiplication:

x 0 1 2
0 0 0 0
1 0 1 2
2 0 2 1

I have to admit these table are a little boring, we can make things more interesting by replacing the numbers by colours. As we are working with modular arithmetic we know that the range of numbers we will come across, lies between 0 and the value we are using for modulus, so we can map these onto some circle of colours. So work mod 151 we get a new table for addition:

Using the same system of colours we can do the same thing for multiplication:

Which is starting to get interesting. We do not need to stop there, we can produce an image where the row number is taken to the power of the column:

This looks a little jumbled, in fact it seems to have very little structure at all. This is not very useful if our goal is to make pretty images, and on this blog that is normally the goal, but it other areas it turns out to be incredibly useful. The process of modular exponentiation is an essential part of public key cryptography, one of the technologies that allows secure communication over the internet. The jumble and lack of pattern that we can see is a sign that modular exponentiation is a good method to use to jumble things up. if there were structure that could be used to help decrypt the messages!

Returning to images, lets make a big version of the multiplication image, mod 1583 (you need to click it to get the full effect, scaling the image down blurs out a lot of structure):

Another option is to make an animation. what happens as we move the modulus value:

There is plenty to study in these images, for example, the curves that can be seen are approximately hyperbolae as they occur when x*y is some fixed value. The central star point occurs in the middle of the image, and there are further stars at 1/3, 2/3, 1/4, 3/4 etc. Can you work out why?

The appearance of hyperbolae perhaps implies that other curves might be possible. What happens if we consider x^2 + y^2? An obvious guess from this formula would be circles and we indeed get (for 151):

Playing around a little further this image comes from x^2 - y^2 +3 x y:

These images are certainly worth repeating for 1583 (again the details get blurred out, so click the images to see the full detail):

To finish let us consider something even simpler. Taking the value of a square to be \frac{x \mod y}{y} this will always give a value between 0 and 1. We can then colour again, and animate with \frac{x \mod Q y}{y} and Q going from 5 to 0:

I first came across these patterns in the December Issue of notices of the AMS, I have always been surprised how little they have been explored. This post is my attempt to do a little to correct that.

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Posted by gelada

Arrange whatever pieces come your way

May 14, 2011

(with apologies to Virginia Wolff)

A simple, classic puzzle is to give two shapes and ask if there is a way to cut one up so the pieces can be rearranged into the other. This game might seem to become silly if both shapes are the same;  if we insist that the new arrangement must be different the game becomes interesting again. Think about it, can you come up with ways to cut up a square so that the pieces can be formed into two different squares? Here is an example, not with a square, but with a rhombus:Having the same shape has an advantage. Think about the letter p below, it is part of the blue trapezium, when we rearrange the tiles the p moves with the shape. As the two shapes are the same we can think of this new p within the original rhomb. We can now repeat the process as many times as we want. In this case, it might be a little unsatisfying, however, as the next step for our p would cut it into two different pieces, as it lies on the edge. So where is it safe to put a p so that it will never get cut up? To answer this we have to follow the cutting lines, and a beautiful pattern emerges:The p would be safe within any of the pentagons, but if it crosses any of the edes it will, eventually be cut apart.

Puzzle: Can you work out the difference between the green and the blue pentagons? (Hint: it relates to the dotted and solid lines in the earlier pictures).

Studying what happens when we can move points or objects around in a space (in this case moving p around a rhomb) is studied in a part of mathematics called Dynamical systems the particular example here is called a Piecewise Isometry  (see this paper for a more formal account of their history and study). I have studied these systems myself, and recently submitted a paper looking at the behaviour and number theory that occurs within the pentagon generating system shown above (take a look! It has lots of pictures as well as more formal mathematics).

As you might have guessed from my preoccupations part of my interest in these systems is the pretty images that they produce; this system is particularly rich. This leads to the image at the top. You can take any rhombus and cut it up in a similar way. Take any rhomb (as shown below) and rotate until the side of the rhomb lines up with the top. This will leave a triangle and a trapezium that can be moved back on top of the original rhomb:Additionally this gives a system where the rotation on the two parts is the same, just around different points. You have to be a little careful, but you can use this to give a system for any angles. For any of these systems we can ask the question: Where is it safe to write p? Every angle gives a different pattern, and tiny changes in the angle leads to large changes in the pattern, however the patterns do relate to one another in some ways, as you can see in this video:

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Posted by gelada

Islamic Geometry

December 20, 2010

Marc Pelletier is a geometric artist, one of the visionaries behind the amazing Zometool system and the designer and builder of 120-cell models including one given to John Conway at Princeton  and one at the Fields institute (given on the occasion of Coxeter’s 95th birthday). More recently he has been working on Islamic tiling patterns, drawing on the work of Jay Bonner, an expert on the geometric art of the Middle East. Marc has created an elegant and general system to generate such tilings with fine control over the symmetry and structures that come out. Here are a couple of sample designs. Marc, Chaim Goodman-Strauss and I were discussing the methods and how they can be put into a mathematical framework.

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Posted by gelada

The strange quest: Mathematics as Concrete Art

October 10, 2009

I have to confess that this post has not been an easy one to write. I wanted to express some ideas that are difficult to put into words. The central, rather playful, thesis is that pure mathematics itself is a branch of concrete art.

Let me begin with some easy facts. This month, I have had the great fortune to be able to take part in a studio exhibition with a group of constructive/concrete artists, including members of the systems group from the 1960′s.  The exhibition was curated by Trevor Clarke in Peter Lowe’s studio.  As a result I have had a chance to have some fascinating conversations with several artists, including Peter Lowe, Trevor Clarke and Jeffrey Steele.

IMG_0131

Spirograph by Richard Grimes

One goal of the exhibition is to start dialogues between artists and technicians, in the spirit of the studio exhibitions that started the systems group in the 1950′s around Adrian Heath and Kenneth and Mary Martin.  With that in mind I would like to give some of the ideas that emerged for me from the conversations.

Constructive and Concrete art arose from a natural conclusion of the process of abstraction. In the case of concrete art this is explicit and stated in Van Doesberg’s “Manifesto of Concrete Art”. Abstraction began by cutting away the figurative and symbolic content of artworks. As this program progresses more and more is cut away until, in a natural conclusion, one is left with nothing. Nothing is a fascinating concept. It is certainly not a trivial one, as we see with relatively late arrival of zero as a number. It does not, however, give a large space in which ideas can work. An empty canvas is an empty canvas and one ends up unable to tell the profound from the lazy. Concrete art emerges from this vacuum as the attempt to produce artworks that are not empty but have no figurative or symbolic meaning. It seems that this goal can be achieved in two distinct ways. One can either take the subconscious or irrational approach that leads to mysticism or the hyper-rational approach to create small works with their own logic.  For obvious reasons I want to consider the second here.

This would seem to argue for a very subjective art, as we must not only consider different personal opinions about a piece, but the individual world that each piece inhabits. Constructivism is more ambitious than this. The idea of removing figure and symbol is not nihilism, but a desire to address raw or objective beauty. It is of course fully accepted that no such beauty exists. This leads to a strange quest, where the goal is known to be unobtainable.

Being interviewed by Peter Lowe about hyperbolic geometry.

Being interviewed by Peter Lowe about hyperbolic geometry.

I come into this from a different point of view. My art does not contain mathematics in order to have no content, but to communicate mathematics. The mathematics is precisely the symbolic meaning. Yet what is mathematics? My personal definition is that mathematics is any concept that can be considered without reference to the real world. I know that this is an intellectual land grab, but I favour overlapping disciplines anyway. Putting this definition together with the constructivist quest for beauty led to some interesting similarities. Let us consider a parallel history of the two topics.

In the late nineteenth century, while painting was starting the move to abstraction with the work of impressionists and others, mathematics was starting a re-examination of its axiomatic roots. Just as art became more abstract the concepts and fields of mathematics were being cut back to rest on top of the set theory of Cantor and Dedekind.  By the 1930′s the impossibilities inherent in both quests were becoming apparent. A year after Van Doesberg published the “Manifesto of Concrete Art”, Göodel published “On formally undecidable propositions of Principia Mathematica and related systems”.  This work showed that whatever axioms one considered (that allow arithmetic) there would always be holes, statements that the axioms did not say were true or false, and one could never be sure that there was not a contradiction a statement both true and false. This was the end of the dream of a perfect mathematical machine. Pure mathematics thus joined in the strange quest, seeking patterns and structure without the possibility of obtaining a final goal.

Work by Gary Woodley

Work by Gary Woodley

In fact by the 1940′s the two subjects were recognising their similarities. Hardy published “A mathematician’s Apology” in 1940 that claimed that mathematics was an art form. With the humility that only a Cambridge academic can feel for his own place in the world he declared:

“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

The quest of a mathematician, to Hardy, was to find beauty and truth, yet without defining exactly what he meant by either. This bears a striking similarity to the vision of constructivism that I described above.  It is no surprise therefore that, perhaps unaware that mathematics had been declared an art, in 1949 Max Bill considered “The Mathematical Approach in Contemporary Art”.

I want to reverse Bill and consider that perhaps the mathematical structure itself, from gauge theory to groups, from motives to matrices from the games of Conway to the technical depth of Grothendieck, stopping on the way to take in the Hopf fibration and bifurcation, the Penrose tiling, and the 57-cell, is simply one giant work of concrete art put together by a cast of thousands.  An edifice built with some logical consistency on the Zermelo-Frankael axioms and the fudge factor axiom of choice.

So here’s to everyone pursuing the strange quest in the belief that the universe has an inexhaustible supply of secrets, and there will always be new beauty to be found even in some of its simplest corners.

Works by Trevor Clarke and John Bremner

Works by Trevor Clarke and John Bremner

The show

A studio presentation linking a selection of historical and contemporary autonomous works with a focus on modular investigations including:

Alexander Rodchenko*
Anthony Hill
Dirk Verhaegen
Edmund Harriss
Freddy Van Parys
Gary Woodley
Getulio Alviani
Jean Spencer
John Bremner
Kenneth Martin
Mary Martin
Peter Lowe
Richard Grimes
Trevor Clarke

Curated by Trevor Clarke in response to an invitation from Peter Lowe to stage a studio exhibition.

*reconstructions

10 Comments | Art, General, Main Article, Mathematics | Tagged: , , , , , , , , , , , , , , | Permalink
Posted by gelada

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