Tag Archives: sprouting stars

Root Routes

by in Chaos Game, Fractals, Geometry, Mathematics, Square Roots, Squares, Triangles and tagged , , , , , , , , , , , , , , | Leave a comment

Suppose a point traces all possible routes jumping half-way towards the three vertices of an equilateral triangle. A special kind of shape appears — a fractal called the Sierpiński triangle that contains copies of itself at smaller and smaller scales:

Sierpiński triangle, jump = 1/2

And what if the point jumps 2/3rds of the way towards the vertices as it traces all possible routes? You get this dull fractal:

Triangle, jump = 2/3

But if you add targets midway along each side of the triangle, you get this fractal with the 2/3rds jump:

Triangle, jump = 2/3, side-targets

Now try the 1/2-jump triangle with a target at the center of the triangle:

Triangle, jump = 1/2, central target

And the 2/3-jump triangle with side-targets and a central target:

Triangle, jump = 2/3, side-targets, central target

But why stop at simple jumps like 1/2 and 2/3? Let’s take the distance to the target, td, and use the function 1-(sqrt(td/7r)), where sqrt() is the square-root and 7r is 7 times the radius of the circumscribing circle:

Triangle, jump = 1-(sqrt(td/7r))

Here’s the same jump with a central target:

Triangle, jump = 1-(sqrt(td/7r)), central target

Now let’s try squares with various kinds of jump. A square with a 1/2-jump fills evenly with points:

Square, jump = 1/2 (animated)

The 2/3-jump does better with a central target:

Square, jump = 2/3, central target

Or with side-targets:

Square, jump = 2/3, side-targets

Now try some more complicated jumps:

Square, jump = 1-sqrt(td/7r)

Square, jump = 1-sqrt(td/15r), side-targets

And what if you ban the point from jumping twice or more towards the same target? You get this fractal:

Square, jump = 1-sqrt(td/6r), ban = prev+0

Now try a ban on jumping towards the target two places clockwise of the previous target:

Square, jump = 1-sqrt(td/6r), ban = prev+2

And the two-place ban with a central target:

Square, jump = 1-sqrt(td/6r), ban = prev+2, central target

And so on:

Square, jump = 1-sqrt(td/6.93r), ban = prev+2, central target

Square, jump = 1-sqrt(td/7r), ban = prev+2, central target

These fractals take account of the previous jump and the pre-previous jump:

Square, jump = 1-sqrt(td/4r), ban = prev+2,2, central target

Square, jump = 1-sqrt(td/5r), ban = prev+2,2, central target

Square, jump = 1-sqrt(td/6r), ban = prev+2,2, central target

Elsewhere other-accessible

• Boole(b)an #2 — fractals created in similar ways

This Charming Dis-Arming

by in Chaos Game, Fractals, Geometry, Mathematics, Squares, Triangles and tagged , , , , , , , , , , , , , , , , , , | Leave a comment

One of the charms of living in an old town or city is finding new routes to familiar places. It’s also one of the charms of maths. Suppose a three-armed star sprouts three half-sized arms from the end of each of its three arms. And then sprouts three quarter-sized arms from the end of each of its nine new arms. And so on. This is what happens:

Three-armed star

3-Star sprouts more arms

Sprouting 3-Star #3

Sprouting 3-Star #4

Sprouting 3-Star #5

Sprouting 3-Star #6

Sprouting 3-Star #7

Sprouting 3-Star #8

Sprouting 3-Star #9

Sprouting 3-Star #10

Sprouting 3-Star #11 — the Sierpiński triangle

Sprouting 3-star (animated)

The final stage is a famous fractal called the Sierpiński triangle — the sprouting 3-star is a new route to a familiar place. But what happens when you trying sprouting a four-armed star in the same way? This does:

Four-armed star #1

Sprouting 4-Star #2

Sprouting 4-Star #3

Sprouting 4-Star #4

Sprouting 4-Star #5

Sprouting 4-Star #6

Sprouting 4-Star #7

Sprouting 4-Star #8

Sprouting 4-Star #9

Sprouting 4-Star #10

Sprouting 4-star (animated)

There’s no obvious fractal with a sprouting 4-star. Not unless you dis-arm the 4-star in some way. For example, you can ban any new arm sprouting in the same direction as the previous arm:

Dis-armed 4-star (+0) #1

Dis-armed 4-Star (+0) #2

Dis-armed 4-Star (+0) #3

Dis-armed 4-Star (+0) #4

Dis-armed 4-Star (+0) #5

Dis-armed 4-Star (+0) #6

Dis-armed 4-Star (+0) #7

Dis-armed 4-Star (+0) #8

Dis-armed 4-Star (+0) #9

Dis-armed 4-Star (+0) #10

Dis-armed 4-star (+0) (animated)

Once again, it’s a new route to a familiar place (for keyly committed core components of the Overlord-of-the-Über-Feral community, anyway). Now try banning an arm sprouting one place clockwise of the previous arm:

Dis-armed 4-Star (+1) #1

Dis-armed 4-Star (+1) #2

Dis-armed 4-Star (+1) #3

Dis-armed 4-Star (+1) #4

Dis-armed 4-Star (+1) #5

Dis-armed 4-Star (+1) #6

Dis-armed 4-Star (+1) #7

Dis-armed 4-Star (+1) #8

Dis-armed 4-Star (+1) #9

Dis-armed 4-Star (+1) #10

Dis-armed 4-Star (+1) (animated)

Again it’s a new route to a familiar place. Now trying banning an arm sprouting two places clockwise (or anti-clockwise) of the previous arm:

Dis-armed 4-Star (+2) #1

Dis-armed 4-Star (+2) #2

Dis-armed 4-Star (+2) #3

Dis-armed 4-Star (+2) #4

Dis-armed 4-Star (+2) #5

Dis-armed 4-Star (+2) #6

Dis-armed 4-Star (+2) #7

Dis-armed 4-Star (+2) #8

Dis-armed 4-Star (+2) #9

Dis-armed 4-Star (+2) #10

Dis-armed 4-Star (+2) (animated)

Once again it’s a new route to a familiar place. And what happens if you ban an arm sprouting three places clockwise (or one place anti-clockwise) of the previous arm? You get a mirror image of the Dis-armed 4-Star (+1):

Dis-armed 4-Star (+3)

Here’s the Dis-armed 4-Star (+1) for comparison:

Dis-armed 4-Star (+1)

Elsewhere other-accessible

• Boole(b)an #2 — other routes to the fractals seen above