4 dimensional genetic fractals

hyper flower

Hyper dimensional flower

This article was taken without edit from my philosophical blog: geneticfractals.wordpress.org and reproduced here for some of its technical relevance. The original article was published on 21 June 2013.


In my last post I boldly claimed to take you by the hand for a stroll in the fourth dimension.  Judging by the high number of views, but not so many likes, I suspect that a healthy degree of skepticism has won the day. And that is how it should be.

I’m the first to agree that in this blog I will drag you into forests of confusion just to show you an exotic flower. But this post in which I explained that we live in a world of fractal dimensions, as in 2.78 or 3.54 rather than just 3; where I suggest that your eyes don’t actually see three dimensions but a little less and where I suggested a DIY experiment for seeing into the fourth dimension, in that post, actually, I was being coldly serious.

Therefore, I feel compelled to get a little more techie on you and present a few dimensional nuts and bolts. I will try to avoid turning this into a lecture; you are here for light and pleasant reading after all.

The whole notion of dimensions is a little confusing and for much of our human existence we have lived quite happily without it. In fact, it is only in recent centuries that artists have painted images with convincing three dimensional perspectives and shadows. This wasn’t just because they didn’t how to do so before (which is true to some degree), it is also because people looking at those paintings didn’t necessarily know how to interpret those three dimensional perspectives. Our ability to see and process such images in our brains has evolved too.

The same is not true for my cat. If I could get my cat to focus on a photograph of a good looking kitty, he would sadly not see that pin-up kitty. He would see a mess of colours on a  piece of paper. His brain cannot translate those colours into an image – let alone  a three dimensional image.

Our brains are key to all this dimensionalist stuff.

Let’s test our brain. What do you see below?

bouncingA bouncing ball, right? Yep, that is what it is. A red ball bouncing up and down along an imaginary line, i,e, in 1 dimension. It is a relatively complex movement that mathematicians call sinusoidal. But since we are clever, we can turn the camera over and look from the side, we see a much simpler perspective of that same moving ball below: a simple rotation in two dimensions: same ball, different perspective.

circularSo, dimensions are all about perspective. The same thing may look very different depending on the angle you are looking from.

Enough of the preschool animations. Bring on the genetic fractals 🙂 Here is one I made earlier. The lighting and rotation give an illusion of three dimensions, which is just fine. Don’t stare too long as you might be lulled into sleep.

gf rotate

So how would we see the fourth dimension if we bumped into it? Let’s do a little mind experiment and pretend we are Einstein, or Zweistein even. Consider a square piece of cardboard. Paint it pink or blue or pick a neutral colour. Now imagine 6 of those and with some scotch you make a box out of this. This requires some dexterity but since we are doing this in our mind, you should manage without wasting too much tape. Now just for fun, we’ll paint the different side of the box with different colours or stick photos on them. Got it?

Now go and play with the first piece of cardboard and the box. You will quickly tire of turning the cardboard over whereas the colourful cube can keep you busy until tea time. Did you enjoy this?

Now, in the same way that the third dimension (the box) is a lot more interesting than the second dimension (the cardboard), the fourth dimension is hugely more interesting than the third. In the fourth dimension you can turn things over in so many ways that you’ll get dizzy. Do you know what happens when you rotate a normal three dimensional thing into the fourth dimension? Check this out:

rotate 4D

How weird is that? Although you are turning this genetic fractal to you hearts content, it looks like it is flattening and expanding. But not turning. This is because one of its three dimension is turning into the fourth dimension and disappears from our view. But it hasn’t deformed, it is exactly as it was. Only our perspective has changed.

Now ask yourself: have you ever seen a thing moving in this way?

I bet you have. This expanding and contracting is quite a common thing. At our sunday market here, there is a french guy with a beret  who plays the accordion. But do you think that he knows that every time he compresses his accordion, that it partially disappears in the fourth dimension? I bet you he doesn’t. He is more interested in the coins that drop from heaven. He is quite good, actually.

Okay, I can tell you are not convinced. You were happy enough to accept that a bouncing red ball was actually a rotating ball even though you couldn’t see that from your perspective. Fair enough, they don’t teach the fourth dimension at preschool yet.

Just to test the limits of your patience. Below are two genetic fractals as before but at a straight angle with each other. One starts with one dimension in the fourth dimension and the other starts in our worldly three dimensions.

two GFs 4D

Now, don’t try to work out what is going on here: the takeaway is that there are two static objects rotating in four dimensions. For three-dimensionalists such as ourselves, it looks like stuff is expanding and disappearing but there is nothing of the sort, these are just rotating objects.

Here is the cracker: whenever you see things in three dimensions that make smooth contractions and expansions in various directions, you have a pretty good chance that you are observing a four dimensional rotation!

I mentioned starling murmurations in my previous post. I was a little cheeky because these are more like fifth or sixth dimensional rotations of a static formation of birds, but I didn’t want to over indulge in your patience.

Other examples are balloons filled with water and tossed in the air. The wavy bulges you see are four dimensional rotations.  Waves themselves are four (or higher) dimensional rotations…

If you are a physicist, you’ll either click unfollow now or you might be heading for the comment field. Hold your horses! Yes, I know that in these examples the fourth dimension is non-spatial and might be water pressure or momentum. But ask yourself, have you ever thought of them as a four dimensional rotations? And when you do, how does that change your view of an object when this view tells you that water pressure, for example, is one of the basic dimensions of an object and not just an additional parameter?

Now, go to the comment box.

Okay, I suspect I have confused you enough.

Just one more comment on my “walk past the tree experiment” from my previous post. This really concerns a fourth spatial dimension. Whenever you are able to find a situation where you can see an object from more than one side at once (without using mirrors), you actually have a four dimensional viewpoint. The “walk past the tree experiment” provides this by combining visual ghost images, (after images) so I’m going to stick with my far fetched claims 🙂

I just hope that someone can replicate this experiment or else, I will face reputational ruin in the face of those I hold in high esteem and the NSA.

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