How many dimension are really there?

In part one of these musings about the meaning of the word dimension, I touched upon the relation between geometry and mathematics. Physical experiences and abstract imaginations had a role in establishing today's usage of the term and in order to think clearly, we should intelligently discriminate theses influences.

However, this task gets complicated further by the introduction of various additional dimensions. That's why I want to address them in this section before I get to psychological and cosmological conclusions in part three.

In the first part I  mentioned the three spacial dimension as a right angled coordinate system, as suggested in Euclids Elements. Toward the end of that text I also brought up none-Euclidean geometries and indirectly hinted, that the angle between them may vary, as long it is not zero (or another multiple of 180). All a dimension has to fulfill, is bringing something new. With one dimension (meaning axis), all possible points are in a line. The second dimension has to be not parallel to the first. Only then, a combination of both opens a plane. And with the addition of the third, which has to leave our plane just with a little, we can get to any point in space, just by moving parallel to these three axis.

To be consistent, the fourth dimension has to lead beyond the space, observable by the everyday eye and add something new. That already detaches it from the source meaning of the word: something measurable by a ruler. As Roger Penrose laid out so eloquently: the shape of quasi-crystal grids indicates, that there might exist a fourth spacial dimension. But unfortunately there is no academically accepted proof for such dimension and neither for a human faculty to percieve it.

Constructing it is easy in theory, you just follow the known rules one step further. As I previously repeated the reasoning of Bernhard Riemann (founder of the modern mathematical view of space): you can divide a dot (1 vertex) and move both dots in opposite directions. The trace of that movement will be the one dimensional line. Our dots confine its shape, be setting a beginning and an end (2 vertexes, 1 line). When you move two overlaying lines in a similar manner you create a two dimensional area (square with 4 vertexes and 4 edges). The third operation in that fashion generates a cube. It has again twice as many vertexes: 4 * 2 = 8, edges  * 2 + number of vertex = 12 and the number of surfaces follows the  same rule. We had previously one square surface, times two = 2 + number of edges (4) = 6. Number of dimensions (3) times 2  = 6 works too. That is because every manyfold (line, square, cube and so on) has to be bordered on any dimension by two manyfolds with a one less dimension. (A line has 2 confining vertexes, square 4 edges and a cube has 6 surfaces).

Now moving two cubes you should get a hypercube, that consist of 16 vertexes, 32 edges, 24 surfaces and 8 bordering cubes. Drawing it on paper is still easy. Just construct a small cube inside of a much bigger one (both share the same center) and then connect the left front lower corner of the small with the left front lower corner of the big and so on. After connecting all eight corner,  you see a projection of a hypercube onto two dimensions (2D).

But imagining a hypercube (or tesseract as coined by Hinton) is to hard to many. What might help is to construct a cube in an animation program. Our small animation just shows a cube not moving for some time. The two cubes we had before is now our one cube at the first frame and at the last time stamp. Track a corner through time and train yourself to see it as an edge of the hypercube. If you get a sense of this, advance to see the movement of a line through time as a surface, till you hopefully reach the time travelling cube, that gets its corners at start and end connected in the same manner as described above in the pen and paper exercise.

As you draw an opaque  cube on a flat medium, parts of its surface get invisible. That is our daily experience, since the retina of an eye has a 2D surface to recognise the outside. But because we have two eyes and there fore two very similar but not identical 2D impressions of the world, our brain renders a sneak peak into full 3D which is good enough for us to orient and act. To get a full 3D image, one would need more than a translucid hologram, but all perspectives at once, like in Ezekiel's description of his vision of God. He saw a fiery cloud, surrounded by four beings, each with four faces steadily directed in the same four directions (front, back, left and right). He gave a detailed description of all faces without any indications that he moved. Such a view requires a 4D perspective, like a complete view of a picture needs a standpoint above it in 3D. Imagine you live inside the plane of a painting. Just your surroundings would be visible.

During the turn of the 19th. century Edwin Abbott Abbott's book "Flatland" became very popular. He delved into thoughts as just described. He also tried to explain spiritual appearings like vanishing ghosts with a movement into the fourth dimension, similar to an object moving above a plane. It would become invisible to creatures that only live inside the a level and can not imagine a third dimension (hence the name flatland).

Such  speculations were not uncommon, even among physicist until Albert Einstein's views got accepted. With the advent of his general relativity theory time became THE fourth dimension. Unified with the three spacial it became an  continuum named Spacetime (like in our animation software exercise). Spacetime is none-Euclidean, since it could be bent by mass. Einstein went even further in his special relativity and abandoned an objective and singular coordinate system all together. Now every mass had its own coordinate system, that changed depending on its speed.

This is not only confusing to laymen, but also divided particle physics from the physics of visible things. One speculative, but somewhat accepted attempt to unify both is Superstring theory, which talks about 22 or more dimensions. Beside the formulas of very advanced math, which brought Edward Witten the  prestigious Fields-Medal, this amount of dimension is simply too much for any visual understanding. Most people get even headaches when seeing the projections of 4D solid on a computer screen. Dimensions at this point are just a fancy term for a variables.

In Euclidean space you need a position (a number) on all three dimension (axis) to define a point. That is why we write all three numbers together as a unit and call it a vector. Changing one value means just moving along one axis and not changing the positions along the others. And adding a dimension means just to add another number to the vector. This is why mathematicians have no trouble to deal with 22 dimension. They mean 22 variables they can change independently, which together describe one object.

And since math is used in all sciences for various applications, the terms floats even further from geometry. Take for instance the five dimensional standard model of personality in psychology. The correlation between the big five traits are not zero. This means you can not just move along one axis without moving somewhat in other. That would be avery weird space to walk inside. That didn't stop scientists to use the word in that context. I would like to see a more nuanced wording, that would differentiate the literal geometric meaning from such metaphoric usage.

But back to the superstrings. One argument why we do not witness all the 22 proclaimed dimensions is: most are rolled up. So these dimension have not just bent axis but the axis are actually circles. This is not a new idea. In fact the very first none-Euclidean geometries (called Hyperbolic or Lobachevskian) have circular dimensions.  A very popular example would be robotics, where drones rotate around their axis. Their direction is (for practical reasons) unified with their position into one four dimensional vector and any movement can be expressed with one mathematical operation.

A more every day example is the HSL or HSV color space, as used in paint programs. Movement along one axis (the third L or V) makes the current color brighter or darker.  The second linear axis (Saturation) is for adding color. At zero you got a grey tone between black and white (dependent on the third value). The greater your position along the second dimension, the more saturated (less grey) the color. The first (Hue) dimension resembles travelling a  rainbow. You go from red to orange, yellow, green,  turquoise, blue, violet and back to red in a circular movement, surrounding the black/grey/white pillar in the middle. This means, that adding 360 degree to the hue value gives you the same positions in this color space. And there are no negative saturation values, since every S - value is a movement from the grey center pillar. Not all colors in this space are visible or even displayable and there are also biological and cultural reasons why this is a simplification (a model). The ability to distinguish (discriminate) where a model is helpful and where not is the beginning of wisdom.

But there is a dimension even stranger (as pointed out in the 1970'ies by Benoit B. Mandelbrot, when he coined the term: the fractal dimension (others call it Hausdorff dimension). Latin ''fractus'' means broken or fragmented and it is pointing to the observation, that most shapes in nature display more and more details or can be broken down to more and more details, the closer you look. Fractals are mathematically generated forms with such a property. Because of their beauty, tracking shots of them are well known, where the camera seems to zoom in endlessly, presenting a never ending plethora of rich patterns and colors, composed by the ever same elements. These are movements along the fractal dimension. Something similar (into the other direction) you see at the beginning of the movie Contact. Zooming away from earth along the planets of our solar system to ever greater star clusters, the milky way, galaxy groups... into the main characters eye.  

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