# Fractal Recipes for Programmable Matter, Did Carol Marcus know the Recipe?

Does the generation of set resolution fractal objects offer an insight as to how future programmable matter may be manipulated? Once again Star Trek may have foreshadowed what may be coming.

# Iterated Function Systems

In a previous article examining the possible relationship between programmable matter as described in Star Trek: Discovery and the underlying nature of spacetime as a continually rewriting hypergraph structure I volunteered that it reminded me of iterated function systems (IFS) as method of constructing fractal objects.

Briefly, and loosely with the mathematical definitions so we don’t get too bogged down, IFS are a method of constructing fractal objects using a set of contraction mappings on a complete metric space¹.

It’s iterated as the set of functions is applied iteratively — simply supplying a point, letting the transformation move it, and using the resultant point as the start for the next iteration. Think like a feedback loop.

If the set of functions is contractive then there is a unique fixed point where the result of the calculated contractions will ultimately converge. This is referred to as the attractor. Interestingly this happens with any number of dimensions, but more usually they’re shown either in 2D or 3D so we don’t have too much trouble in understanding the visualisation!

(There are examples of non-contractive IFS converging to attractors but that’s really beyond the scope of this article and not what I’m interested in talking about.)

A common fractal object easily constructed with an IFS would be the Sierpinski Triangle whereby we start with a single triangle, make three half size copies, and move them to cover the corners original triangle. This process, described mathematically as a set of affine transformations for the purposes of an IFS, is then repeated.

This process can be continued indefinitely, resulting in smaller and smaller triangles, but naturally for the purposes of display we need only render to the available resolution of the display device on which it is viewed.

The Sierpinski Gasket, or Tetrahedron as it is called below, is similarly described but easily rendered in higher dimensions.

Any set of equations as long as they are contractive can be used². For the Sierpinksi Triangle three similar ones were used, but there’s nothing stopping us from using more and they could be different.

The Barnsley Fern, is a common example, using four equations with differing coefficients, that converge to a a fractal object very similar to the black spleenwort fern.

To construct:

A starting point is chosen randomly anywhere on the 2D plane.

A random equation, from the four originally specified, is chosen and the point transformed as described.

The new point is fed back into the system.

Ultimately, after many iterations, the system converges to the image on the left.

# Simple Equations, Complex Objects³

As you can imagine an infinite number of images may be constructed using an arbitrary number of equations and equally an arbitrary selection of their coefficients. In any number of dimensions.

**Generating Rules from Objects**

In the past, research was done on deriving the equations to represent real world objects, with some success, and was known as fractal image compression. This lossy method, rendered to a specified resolution, was able to generate an attractor similar to the offered original image.

By deriving more accurate coefficients and spreading the equations out across portions of the image better accuracy was achieved, but at the cost of both the processing involved in deriving the source equations but also the storage of the equations themselves. Higher orders of equations, with more coefficients, would require more storage.

I worked in this area for a time and this was, in fact, the area of my own PhD.

**Generating Objects from Rules**

Now, consider the Wolfram Physics Project’s representation of spacetime as a continually rewriting hypergraph — is it possible that a specified hypergraph could be recreated from a set of original equations?

Well, yes, of course, as their representation of spacetime as a hypergraph depends directly upon a set of rules that dictate how the hypergraph evolves.

This is what got me to thinking of IFS in my previous article.

It may then be possible to recreate a desired, specific, state of the hypergraph to represent any object⁴ in spacetime given a particular set of rewriting rules.

Not only that but a hypergraph may be newly created and inserted directly into existing space, existing spacetime could be replaced by a new hypergraph, or even existing spacetime could be modified into a new hypergraph.

**Insert**

Perhaps the creation of a new hypergraph may be the construction of a real world mythical bag of holding containing anything from a pocket universe to a simple bag of sweets which could then be sewn inside your existing trouser pocket? Going further, perhaps a sympathetic reality programmer would purposefully construct a computer inside this new region of spacetime that could itself construct endless gobstoppers by rewriting its own hypergraph such that you always find one when you reach into your pocket!

**Modify**

The situation depicting the rewriting of existing spacetime into a new hypergraph is reminiscent of the Genesis Device, again from the Star Trek franchise.

From the second, and in my opinion the best of all of the big screen adaptations: Star Trek II: The Wrath of Khan.

“Now, watch out. Here comes Genesis. We’ll do it for you in 6 minutes.”

— Dr. McCoy

This would be akin to purposefully rewriting existing reality. The programmer would have to be very careful such as to take account of the existing arrangement of reality so’s not to accidentally alter anything in it.

At least, not too much.

One way to think of this would be modern day techno-magic, perhaps turning water into wine as it is poured from bottle to glass perhaps? But, it would be no trick or sleight of hand, it would be the tangible alteration of reality.

**Delete**

And perhaps the most ominous action a programmer could take, the deletion of a region of spacetime. Let’s hope there’s an existential hypergraph recycle bin from which accidentally deleted reality can be retrieved.

Or, do we consider Dr McCoy’s thoughts from The Wrath of Khan and go one step further and imagine a war between competing civilisations each endlessly rewriting and deleting each other’s reality.

Could this be something like the Last Great Time War, an existential conflict that may play out between the Time Lords and the Daleks but not only tinkering with time, but with entwined space as well.

A truly frightening prospect.

# Fractally Programmable Spacetime

If we consider that at some future point we may be able to manipulate underlying spacetime directly by interacting with its underlying hypergraph a tantalising range of possibilities opens up.

However, at this ultimate level of detail, there may be many, many vertices and edges before we even begin to consider the complexity of the equations necessary to describe the rewriting system that we need to master.

The computational cost cannot even be estimated at this point should the project begin to approach a convincing description of spacetime.

As Wolfram et al. have demonstrated, fairly simple rewriting equations or systems can describe extremely complex hypergraphs in the same way as a simple set of equations can fully describe the Barnsley Fern.

(Much of the fundamental work here goes back to Stephen Wolfram’s own study of cellular automata as described in his work A New Kind of Science.)

This doesn’t mean to say that simplistic systems can fully describe our current reality’s own objects in the spacetime hypergraph such as an apple pie, a grain of sand, or an electron. But, we can’t say that for sure, at least not yet anyway.

We do tend to think of complex objects requiring complex descriptions but clearly with fractal objects, for example, that’s not the case.

Perhaps it is possible that simple equations similar to those describing the Barnsley Fern applied to the spacetime hypergraph may indeed cause rewrites sufficient to create something as incredibly complex as the mathematical fern but as an object in our actual perceived reality.

We may need exabytes of information to construct an electron, perhaps, or maybe it’s only actually just a few bytes. It does make me wonder what the hypergraph equivalent of complex fractal objects may be, when described by precise, simplistic formulae!

¹ For convenience think of a Metric Space in terms of points in 2D cartesian space, like dots on a plane. The space is the plane and some way of measuring the distance between the points is defined. More information, formally, here. There is, of course, no actual limit to the number of dimensions.

² Arnaud E. Jacquin. “Image Coding Based on a Fractal Theory of Iterated Contractive Image Transformations.” IEEE Transactions on Image Processing, 1(1), 1992.

³ No pun intended, if you’re mathematically minded and are familiar with fractal geometry. Especially with the Mandelbrot Set.

⁴ I say “any object” but as we will be rewriting spacetime itself the full ramifications of this aren’t yet know — all kinds of as yet unknown esoteric objects ,with strange properties, may be created. Let’s not even get into time.