## Mathematical Precursor

I really, really wanted to talk about maths today. I’ve been teaching it for (I kid you not) about 14 hours every weekday for the last few weeks, and it’s been pretty intense.

I feel like I’m starting to get a feeling for what most people find difficult, frustrating and boring about mathematics. I think I’ve also got some pretty awesome insight into the world of maths, how people learn it and how it should be taught.

However, I have 78 (!) new non-spam emails in my inbox and a class to teach very soon. When I’m done, I am going to go home and catch up on my sleep and try not to have a brain haemorrhage. I’ll save my insighty goodness for another day, and instead post some awesome mathematical images that I’ve found for your amusement and delight.

The first is a mathematical diagram representing what is called the projective space PG(3,2), and I think it has a pleasing fiveness to it:

I got this from a paper called “Pretty pictures of Geometries” by B. Polster. He says:

“Why are good pictures important? Two of the main reasons that come to mind are the following:

-To convey some of the abstract beauty of the objects we study to people outside our field. This seems to be especially important today as it becomes more and more important to “justify” and “sell” the kind of research we are fascinated by.

-Many of us think in terms of pictures of various degrees of abstraction. The kind of pictures we want to concentrate on in this note are immediately accessible and can serve to lure students into studying incidence geometry and as a first step in teaching students pictorial thinking in geometry.”

The kinds of mathematics I’m engaged in deals with abstraction as well. I study Algebra, not Geometry, so the beauty I find there is almost always unable to be drawn, though there are a few interesting diagrams. Here is a picture representing the “root system” of the algebra G_2, the object I studied for my Honours thesis:

What it ‘really means’ is rather complex, but for the moment, it’s enough to say “oooh, pleasing sixness! Hey, isn’t that the Star of David?” or something like that.

Here are some more pictures, starting with fractals:

(A fractal called “God’s Own Pentagram”)

(The famous “Mandelbrot Set”

(I don’t know what this one is called, but I like how organic it looks)

Each of these fractals have a property called “Self-similarity”, which means that you can find copies of the whole or parts of the image, in miniature, throughout the fractal. If you look closely, you’ll see the self-similarity in each of those fractals above. You can even download and try this absolutely fantastic fractal zoomer called XaoS, which lets you both zoom in on parts of a fractal to see this self-similarity in action, and form Julia sets like this one:

Here’s how to represent a 4-dimensional object using a 3D image, utilising time as a replacement for a fourth spacial dimension:

I find this rather hypnotic! I’ve heard that those who study geometry get a ‘feeling’ for how things look in 4D space by doing things like this in their brain for all kinds of 4D shapes. Wat.

I have to go to class now, so I’d better cut this short. I’d like to leave off with a question, though. Consider this image:

Do you see her rotating clockwise or anti-clockwise? It’s important to note that nothing is rotating – it’s just a moving 2D silhouette. Our brains interpret the image as rotating, but you can get her to reverse direction if you concentrate hard enough. You have to force your brain to interpret the data it’s getting in a completely different way, and it helps to figure out what parts of the image are making your brain say “this is a rotating woman!”.

Tell me what you see, and if you can reverse her rotation in your brain! Next week, I’ll talk more about maths.

## juliadactyl said,

February 13, 2009 at 5:20 pm

I can make her switch rotations. It’s rad.

I love your picturemaths, guy. Especially the Julia fractals, because this one time during the times I googleimagesearched “Julia” and that’s what came up and it totally blew my mind.

Also, that pleasing fiveness you have up there is very pleasing, just like the flower from that day. Thank you for these! I look forward to reading more about your mathsings, also I have some Epic Plans for chillaxing when all your works are done.

## Danoot said,

February 13, 2009 at 5:28 pm

I used to be able to switch this picture pretty much at will.

now it is stuck in clockwise.

why you gonna be a hater, brain? I demand to control my perceptions!

## misterfinn said,

February 13, 2009 at 10:55 pm

It took me a while, but I could eventually make her switch. I would like to understand more about maths, since I’ve always had a massive problem with perception and it saddens me that I can’t make the same connections with these kinds of things other people can.

## chromefist said,

February 15, 2009 at 12:54 am

Yep, I am also trapped in clockwise. I think maybe I could do it once, but I can’t remember for sure.

Also, 4 spatial dimensions is trippy. Replacing one with time makes it pretty; but doesn’t soothe my troubled brane. Stop tinkering with the fabric of reality, k?

## Heather said,

February 15, 2009 at 11:21 am

I primarily get clockwise. I tried looking at her limbs to see whether I could make them change direction, because that was what (it seemed to me) was determining how I perceived her direction, but that did me no good, perhaps because they stick out a bunch it is harder for me to interpret the motion in another way.. I managed to get it to change direction last night, I think by staring at the boob part of the silhouette. Yeah, classy. I can’t do it now, though, and even when I managed to make it go anticlockwise I wasn’t able to switch back and forth at will.

## chromefist said,

February 15, 2009 at 11:13 pm

Also: The extended foot shadow, which only appears when her leg is moving right to left seems to strongly suggest it’s spinning clockwise. Although I suppose the floor could be tilted sharply to produce that effect while going anti-clockwise.

## ZeroSum Ruler said,

October 19, 2010 at 12:09 am

I’ve been searching for years (on and off, of course) for a simple way to teach algebra students about fractals. My students know about imaginary numbers, which, in my limited understanding of fractals, I read are part of creating fractals. If there an algebra activity anywhere that kids can do to create a fractal- albeit simple!- on, say, graph paper?