### Foreword

Following my last post on the “

*…first, second, and third dimensions, and why fractals don’t belong to any of them…*“, this post is about documenting my journey as I delve deeper into the subject of**fractals, mathematics, andÂ geometry**.The study of fractals is an intensely vast topic. So much so that I’m convinced you could easily spend several lifetimes studying them. That being said,Â I chose to focus specifically on

**single-curve geometry**.Â But, keep in mind that I’m only really scratching the surface of what there is to explore.# 4.0 Classic Space-Filling

Inspired byÂ Georg Cantor’sÂ research on infinity near the end of the 19th century, mathematicians were interested in finding

**a mapping of a one-dimensional line into two-dimensionalÂ space**– a curve that will pass through through every single point in a given space.Jeffrey VentrellaÂ writes thatÂ “a space-filling curve can be described as a continuous mapping from a lower-dimensional space into a higher-dimensional space.” In other words, an initial one-dimensional curve is developed to

**increase its length and curvature**– the amount of space in occupies in two dimensions. And in the mathematical world, where**a curve technically has no thickness and space isÂ infinitelyÂ vast**, this can be done indefinitely.### 4.1 Early Examples

In 1890, Giuseppe Peano discovered the first of what would be called space-filingÂ curves:

An initial ‘curve’ is drawn, then each element of the curve is replace by the whole thing. Here it is done four times, and it’s easy to imagine how you

**can keep doing this over and over again**. One would think that if you kept doing thisÂ indefinitely,**this one-dimensional curve would eventually fill all ofÂ two-dimensional space**and become a surface. However it can’t, since it technically has no thickness. So it will be as close as you can get to a surface, without actually being aÂ surface (I think.. I’m not that sure..)A year later, David Hilbert followed with his slightly simplerÂ space-filingÂ curve:

In 1904,Â Helge von Koch describes a single complex continuous curve, generated withÂ rudimentary geometry.

Around 1967, NASAÂ physicists John Heighway, Bruce Banks, and William Harter discovered what is now commonly known as the Dragon Curve.

### 4.2 Later Examples

You may have noticed that some of these curves are better at filling space than others, and this is related to their dimensional measure. They fall under the category of fractals because they’re neither one-dimensional, nor two-dimensional, but sit somewhere in between.Â For these examples, their dimension is often defined by exactly

**Â how much space they fill when iterated infinitely**.While these are some of the earliest space-filling curves to be discovered, they are just a handful of the likely

**endless different variations**that are possible.Â Jeffrey Ventrella spent over twenty-five years exploring fractal curves, and has illustrated over 200 hundred of them in his book ‘Brain-Filling Curves, A Fractal Bestiary.’ They are organised according to a taxonomy of fractal curve families, and areÂ shownÂ with aÂ uniqueÂ genetic code.Incidentally, in an attempt to recreate one of the fractals I found in Jeffery Ventrella’s book, I accidentally created a slightly different fractal. As far as I’m concerned, I’ve created a new fractal and amÂ unofficiallyÂ naming it ‘

**Nicolino’s Quatrefoil**.’ The following was created inÂ*Rhino*and*Grasshopper,*in conjunctionÂ*Anemone*.You can find beautifully animated space-filling curves here:

(along with some other great videos by ‘3Blue1Brown’Â discussing the nature of space-filling curves, fractals, infinite math, and more)

#### On A Strange Note:

It’s possible to iterate a version of the Hilbert Curve that (once repeatedÂ infinity) can fill three-dimensional space.

As an object, it seems perplexinglyÂ

**difficult to categorize**.Â It is a single, one-dimensional, curve that is ‘bent’ in space following simple, repeating rules. Following the same logic as the original Hilbert Curve, we know that this can be doneÂ indefinitely, but this time it is transforming into a volume instead of a surface. (Ignoring the fact that it is represented with a thickness) It is a one-dimensionalÂ curveÂ transformingÂ into a three-dimensionalÂ volume, but is never a two-dimensional surface? As you keep iterating it, its dimension gradually increases from 1 to eventually 3, but**will never, ever, ever be 2??**Nevertheless this does actually support a statement I made in my last post suggesting “

*…*“**there is no ‘first’ or ‘second’ dimension.**It’s a bit like pouring three cups of water into a vase and asking someone which cup is the first one. The question doesn’t even make sense…# 5.0 Avant-GardeÂ Space-Filling

In the case of the original space-filling curve, the goal was to fill all of infinite space. However the fundamentalÂ behaviourÂ of these curves change quite drastically when we start to

**play with the rules used to generate them**.Â For starters, they do not have to be so mathematically tidy, or geometrically pure. The following curves can beÂ subdivided infinitely, making them true space-filling curves. But, what makes them special is**the ability to control the space-filling process**, whereas the original space-filling curves offer little to noÂ artisticÂ license.### 5.1 The Traveling Salesman Problem

Let’s say that we change the criteria, from passing through everyÂ singleÂ point in space, to

**passing only through the ones we choose**.Â This now becomes a well documented computational problem that hasÂ immediate ‘real world’ applications.Our figurative traveling salesman wishes to travel theÂ country selling his goods in as many cities as he can. In order to maximize his net profit,

**he must make his journey as short as possible**, while of course still visiting every city on his list. His best possible route becomesÂ exponentiallyÂ moreÂ challengingÂ to work out, as even just a handful of cities can generate thousands of permutations.There are a variety of different strategies to tackle this problem, a few of which are described here:

The result is ultimately

**a single curve, filling a space in a uniquely controlledÂ fashion**. This method can be used to create single-lined drawings based on points extracted fromÂ Voronoi diagrams, a topic explored byÂ Arjan Westerdiep:### 5.2 Differential Growth

If we let physics (rather than math) dictate the growth of the curve, the result becomes

**more organic and less controlled**.In this example

*Rhino*is used with*Grasshopper*and*Kangaroo 2*.Â A curve is drawn on a plain, broken into segments, then gradually increased in length. As long as the curve is not allowed to cross itself (which is achieved here with ‘Collision Spheres’), the result is a curve that is pretty good at**uniformly filling space**.TheÂ geometryÂ doesn’t even have to be bound by a planar surface; It can be done on

**any two-dimensional surface**(or in three-dimensions (even higher spacial dimensions I guess..)).Additionally,

*Anemone*can be used in conjunction with*Kangaroo 2*to continuously subdivideÂ the curve as it grows. The result is much smoother, as well as far more organic.Of course the process can also be reversed, allowing the curve to flow seamlessly from one space to another.

Here are far more complex examples of growth simulations exploring various rules and parameters:

# 6.0 Developing Fractal Curves

In the interest of creating something a little more tangible, it is possible to increase theÂ dimension of these curves.

**Recording the progressive iterations of a space filling curve**allow us to generate what is essentially a space-filling surface. This new surface has the unique quality of being**able to fill a three-dimensional space**of any shape and size,Â while being a single surface. ItÂ of course also shares the same qualities as its source curves, where it keep increasing in surface area (and can do soÂ indefinitely).If you were to keep gradually (butÂ indefinitely) increasing the area of a surface this way in a finite space, the result will be

**a two-dimensional surface seamlessly transforming into a three-dimensional volume**.### 6.1 Dragon’s Feet

Here is an example of turning the dragon curve into a space-filling surface.

**Each iteration is recorded and offset in depth**, all of which inform the generation of a surface that looselyÂ flows through each of them. This was again achieved with*Rhino*and*Grasshopper*.I don’t believe this geometry has a name beyond ‘the developing dragon curve’, so I’ve called it ‘

*Dragon’s Feet*.’Adding a little thickness to the model allow us to 3D print it.

### 6.2 Hilbert’s Curtain

Here is the Hilbert Curve going through the same process, which I am aptly naming ‘

*Hilbert’s Curtain*.’3D Printing Space-Filling Curves with Henry Segerman at Numberphile:

‘Developing Fractal Curves’ byÂ Geoffrey Irving & Henry Segerman:

### 6.3 Developing Whale Curve

Unsurprisingly this can also be done with differentially grown curve. The respective difference being that

**this method fills a specific space in a less controlled manner**.In this caseÂ with

*Kangaroo 2*Â is used to grow a curve into the shape of a whale. Like before, each iteration is used to inform a single-surface geometry.
You really should give credit to Henry Segerman. He did this first.

Hi Nick could you share the grasshopper script of the planar differential growth?