### Foreword

*…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, and geometry**.

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

# 4.0 Classic Space-Filling

**a mapping of a one-dimensional line into two-dimensional space**– a curve that will pass through through every single point in a given space.

**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 is infinitely vast**, this can be done indefinitely.

### 4.1 Early Examples

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

**can keep doing this over and over again**. One would think that if you kept doing this indefinitely,

**this one-dimensional curve would eventually fill all of 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 a surface (I think.. I’m not that sure..)

### 4.2 Later Examples

**how much space they fill when iterated infinitely**.

**endless different variations**that are possible. 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 are shown with a unique genetic code.

**Nicolino’s Quatrefoil**.’ The following was created in

*Rhino*and

*Grasshopper,*in conjunction

*Anemone*.

#### On A Strange Note:

**difficult to categorize**. 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 done 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-dimensional curve transforming into a three-dimensional 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??**

*…*“

**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-Garde Space-Filling

**play with the rules used to generate them**. For starters, they do not have to be so mathematically tidy, or geometrically pure. The following curves can be 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 no artistic license.

### 5.1 The Traveling Salesman Problem

**passing only through the ones we choose**. This now becomes a well documented computational problem that has immediate ‘real world’ applications.

**he must make his journey as short as possible**, while of course still visiting every city on his list. His best possible route becomes exponentially more challenging to work out, as even just a handful of cities can generate thousands of permutations.

**a single curve, filling a space in a uniquely controlled fashion**. This method can be used to create single-lined drawings based on points extracted from Voronoi diagrams, a topic explored by Arjan Westerdiep:

### 5.2 Differential Growth

**more organic and less controlled**.

*Rhino*is used with

*Grasshopper*and

*Kangaroo 2*. 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**.

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

*Anemone*can be used in conjunction with

*Kangaroo 2*to continuously subdivide the curve as it grows. The result is much smoother, as well as far more organic.

# 6.0 Developing Fractal 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, while being a single surface. It of course also shares the same qualities as its source curves, where it keep increasing in surface area (and can do so indefinitely).

**a two-dimensional surface seamlessly transforming into a three-dimensional volume**.

### 6.1 Dragon’s Feet

**Each iteration is recorded and offset in depth**, all of which inform the generation of a surface that loosely flows through each of them. This was again achieved with

*Rhino*and

*Grasshopper*.

*Dragon’s Feet*.’

### 6.2 Hilbert’s Curtain

*Hilbert’s Curtain*.’

### 6.3 Developing Whale Curve

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

*Kangaroo 2*is used to grow a curve into the shape of a whale. Like before, each iteration is used to inform a single-surface geometry.