Hyper fractals for fourth dimension geeks

flower

This is not a coming out; I’ve been known to go on about fourth dimensions and other weird stuff like non-integer dimensions. I am otherwise relatively normal but when it comes to dimensions, I am an incurable geek.

One thing that has frustrated me is that 4-dimensional objects are a little hard to create, manipulate and view. Partly because they are hard to get your head around and partly because the maths gets a little sticky. Even adding just one dimension to a cube to make it a hyper cube, is sufficiently mind-boggling to give up once and for all.

tesseract
Rotating tesseract (hyper cube) – credit Wikimedia

You can imagine my elation when I discovered that the mathematical formulation of the my dear and beloved genetic fractals could be extended to any number of dimensions. The reason, if you really want to know, is that genetic fractals are based on rotations. Starting with a one-dimensional line, I can rotate that around one of its extremities and get a circle in two dimensions. I can rotated that circle and get a sphere. So, by rotating a line twice, I can get to a point anywhere on a sphere. To get to the next dimensions, all I have to do is rotate into a 4th dimension. This is very hard, if not impossible, to imagine but in the model for genetic fractals, this is trivially easy. What’s more, without any effort, we can add as many dimensions as we like without getting our knickers in a twist.

So, four dimensional fractals, or sweeter still, “hyper fractals” are within reach. The other neat thing about genetic fractals is that they are intrinsically organic. Since our brain easily recognizes organic shapes, perhaps, just perhaps, we can get our head around hyper genetic fractals.

A hyper genetic fractal is an object that grows into a certain direction and from time to time it splits into several branches which themselves will split into branches again. Since the direction is now in a four dimensional space, the genetic fractal will grow into a four dimensional object. That’s all it takes. Now, one snag with four dimensional objects is that we can only ever see three of the four dimensions. If we want to get a sense of this four dimensional fractal we need to rotate it in four dimensional space and just look at it from our limited three dimensional perspective. Below is one such fractal seen from different four dimensional viewpoints.

4D fractal before rotation
4D genetic fractal before rotation
4D genetic fractal rotated
4D genetic fractal rotated
4 dimensional fractal rotated further
4 dimensional fractal rotated further

Now, if you are like me then this looks strange. But let me be explicit: these four dimensional images are of the same static object seen from different four dimensional angles.The maths don’t lie.

The reason that they look like they are different is because our brain isn’t able to form a coherent image in four dimensions. Our brains need to evolve still.

We don’t have such a problem with two dimensional projections of three dimensional forms. Even the cubes below without shading are easily reconciled as being the same object from different angles. But, as two dimensional images, they are completely different.

cubes from different angles
cubes from different angles

Is there there any hope that we can get a good grasp of hyper dimensional objects? I’d like to think that the answer is yes. In part because I now have a decent tool for creating hyper fractals and in part because we do have an innate sense of a fourth dimension: time.

My homework for this week is to create an animation of a four dimensional rotation of a hyper fractal. Perhaps that will get us a little further. Check it out here.

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