# Jumper to Jumper

Previously I’ve looked at fractals created by a point moving half-way towards the random chosen vertex of a polygon. But you can also choose an initial vertex, then choose a new vertex by adding a random number to that initial vertex. Then repeat. For example, if the polygon is a square and the initial vertex is v = 1, then choose v + 3 = 4 or v – 1 = 3, and so on.

You can then ban or un-ban the choice of vertex-jump as you can ban or un-ban direct choices of vertex. These two methods of random choice are effectively the same, but one can be simpler to program than the other. That’s why I’ve come across some new fractals by using vertex-jumps. Here they are:

vertices = 4, vertex-jump = (1,2,3,4), ban on same choice twice in a row

vertices = 4, vertex-jump = (1,2,3,4), ban on 2 in row (black-and-white version)

v = 4, vj = (1,2,3,4), ban on choice c + 2 from previous choice c

v = 4, vj = (1,2,3,4), ban c + 2 (animated gif)

vj = (1,2,3,4), ban c + 2 (black-and-white)

vj = (1,2,3,4), ban c + 0 at time t+1 unless c + 0 at time t-1

vj = (1,2,3,4), ban c + 0 at t+1, unless c + 0 at t-1 (black-and-white)

vj = (1,2,3,4,5), ban c + 0

vj = (0,1,2,3,4), ban c + 0

vj = (0,1,2,3,4), ban c + 0 (black-and-white)

vj = (1,2,3,4), ban c + 2 at t+1 unless c + 2 at t-1 (animated gif)

vj = (1,2,3,4), ban c + various at t+1 unless c + various at t-1 (animated gif)

vj = (1,2,3,4,5), ban c + 0 at t+1 unless c + 0 at t-1

vj = (-2,-1,0,1,2), ban c + 0

vj = (-2,-1,0,1,2), ban c + 0 (black-and-white)

vj = (0,1,2,3,4), ban c + va unless c + va

v = 5, vj = (1,2,3,4), ban c + 0

v = 5, vj = (1,2,3,4), ban c + 2

v = 5, vj = (0,1,2,3), ban c + 3

v = 6, vj = (0,1,2,3), ban c + 2

v = 6, vj = va, ban c + va (animated gif)

# M.i.P. Trip

The Latin phrase multum in parvo means “much in little”. It’s a good way of describing the construction of fractals, where the application of very simple rules can produce great complexity and beauty. For example, what could be simpler than dividing a square into smaller squares and discarding some of the smaller squares?

Yet repeated applications of divide-and-discard can produce complexity out of even a 2×2 square. Divide a square into four squares, discard one of the squares, then repeat with the smaller squares, like this:

Increase the sides of the square by a little and you increase the number of fractals by a lot. A 3×3 square yields these fractals:

And the 4×4 and 5×5 fractals yield more:

# The Hex Fractor

A regular hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into three more equilateral triangles and a regular hexagon. If you discard the three triangles and repeat, you create a fractal, like this:

Adjusting the sides of the internal hexagon creates new fractals:

Discarding a hexagon after each subdivision creates new shapes:

And you can start with another regular polygon, divide it into triangles, then proceed with the hexagons:

# Hextra Texture

A hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into a hexagon and three more equilateral triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.

Previously pre-posted (please peruse):