Tag Archives: Sierpinski triangles

Square Routes Re-Re-Re-Re-Re-Re-Revisited

by in Fractals, Geometry, Mathematics, Pentagons, Polygons, Squares, Triangles and tagged , , , , , , , , | Leave a comment

Suppose you trace all possible routes followed by a point inside a triangle jumping halfway towards one or another of the three vertices of the triangle. If you mark each jump, you get a famous geometrical shape called the Sierpiński triangle (or Sierpiński sieve).

Sierpiński triangle found by tracing all possible routes for a point jumping halfway towards the vertices of a triangle

The Sierpiński triangle is a fractal, because it contains copies of itself at smaller and smaller scales. Now try the same thing with a square. If you trace all possible routes followed by a point inside a square jumping halfway towards one or another of the four vertices of the square, you don’t get an obvious fractal. Instead, the interior of the square fills steadily (and will eventually be completely solid):

Routes of a point jumping halfway towards vertices of a square

Try a variant. If the point is banned from jumping towards the same vertex twice or more in a row, the routes trace out a fractal that looks like this:

Ban on choosing same vertex twice or more in a row

If the point is banned from jumping towards the vertex one place anti-clockwise of the vertex it’s just jumped towards, you get a fractal like this:

Ban on jumping towards vertex one place anti-clockwise of previously chosen vertex

And if the point can’t jump towards two places clockwise or anti-clockwise of the currently chosen vertex, this fractal appears (called a T-square fractal):

Ban on jumping towards the vertex diagonally opposite of the previously chosen vertex

That ban is equivalent to banning the point from jumping from the vertex diagonally opposite to the vertex it’s just jumped towards. Finally, here’s the fractal created when you ban the point from jumping towards the vertex one place clockwise of the vertex it’s just jumped towards:

Ban on jumping towards vertex one place clockwise of previously chosen vertex

As you can see, the fractal is a mirror-image of the one-place-anti-clockwise-ban fractal.

I discovered the ban-construction of those fractals more than twenty years ago. Then I found that I was re-discovering the same fractals when I looked at what first seemed like completely different ways of constructing fractals. There are lots of different routes to the same result. I’ve recently discovered yet another route. Let’s try what seems like an entirely different way of constructing fractals. Take a square and erect four new half-sized squares, sq1, sq2, sq3, sq4, on each corner. Then erect three more quarter-sized squares on the outward facing corners of sq1, sq2, sq3 and sq4. Carry on doing that and see what happens at the end when you remove all the previous stages of construction:

Animation of the new construction

Animation in black-and-white

It’s the T-square fractal again. Now try rotating the squares you add at stage 3 and see what happens (the rotation means that two new squares are added on adjacent outward-facing corners and one new square on the inward-facing corner):

Animation of the construction

It’s the one-place-clockwise-ban fractal again. Now try rotating the squares two places, so that two new squares are added on diagonally opposite outward-facing corners and one new square on the inward-facing corner:

Animation of the construction

It’s the same-vertex-ban fractal again. Finally, rotate squares one place more:

Animation of the construction

It’s the one-place-clockwise-ban fractal again. And this method isn’t confined to squares. Here’s what happens when you add 5/8th-sized triangles to the corners of triangles:

Animation of the construction

And here’s what happens when you add 5/13th-sized pentagons to the corners of pentagons:

Animation of the construction

Finally, here’s a variant on that pentagonal fractal (adding two rather than four pentagons at stage 3 and higher):

Animation of the construction

Previously pre-posted (please peruse):

• Square Routes
• Square Routes Revisited
• Square Routes Re-Revisited
• Square Routes Re-Re-Revisited
• Square Routes Re-Re-Re-Revisited
• Square Routes Re-Re-Re-Re-Revisited
• Square Routes Re-Re-Re-Re-Re-Revisited

We Can Circ It Out

by in Circles, Fractals, Geometry, Mathematics, Polygons, Squares, Triangles and tagged , , , , , , , , , , , , , , | Leave a comment

It’s a pretty little problem to convert this triangular fractal…

Sierpiński triangle (Wikipedia)

…into its circular equivalent:

Sierpiński triangle as circle

Sierpiński triangle to circle (animated)

But once you’ve circ’d it out, as it were, you can easily adapt the technique to fractals based on other polygons:

T-square fractal (Wikipedia)
⇓

T-square fractal as circle

T-square fractal to circle (animated)

Elsewhere other-accessible…

• Dilating the Delta — more on converting polygonic fractals to circles…

See-Saw Jaw

by in Fractals, Geometry, Mathematics and tagged , , , , , , , , , | Leave a comment

From Sierpiński triangle to T-square to Mandibles (and back again) (animated)
(Open in new window if distorted)

Elsewhere other-accessible…

• Mandibular Metamorphosis — explaining the animation above
• Agnathous Analysis — more on the Sierpiński triangle and T-square fractal

Square Routes Re-Re-Re-Re-Re-Revisited

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

For a good example of how more can be less, try the chaos game. You trace a point jumping repeatedly 1/n of the way towards a randomly chosen vertex of a regular polygon. When the polygon is a triangle and 1/n = 1/2, this is what happens:

Chaos triangle #1

Chaos triangle #2

Chaos triangle #3

Chaos triangle #4

Chaos triangle #5

Chaos triangle #6

Chaos triangle #7

As you can see, this simple chaos game creates a fractal known as the Sierpiński triangle (or Sierpiński sieve). Now try more and discover that it’s less. When you play the chaos game with a square, this is what happens:

Chaos square #1

Chaos square #2

Chaos square #3

Chaos square #4

Chaos square #5

Chaos square #6

Chaos square #7

As you can see, more is less: the interior of the square simply fills with points and no attractive fractal appears. And because that was more is less, let’s see how less is more. What happens if you restrict the way in which the point inside the square can jump? Suppose it can’t jump twice towards the same vertex (i.e., the vertex v+0 is banned). This fractal appears:

Ban on choosing vertex [v+0]

And if the point can’t jump towards the vertex one place anti-clockwise of the currently chosen vertex, this fractal appears:

Ban on vertex [v+1] (or [v-1], depending on how you number the vertices)

And if the point can’t jump towards two places clockwise or anti-clockwise of the currently chosen vertex, this fractal appears:

Ban on vertex [v+2], i.e. the diagonally opposite vertex

At least, that is one possible route to those three particular fractals. You see another route, start with this simple fractal, where dividing and discarding parts of a square creates a Sierpiński triangle:

Square to Sierpiński triangle #1

Square to Sierpiński triangle #2

Square to Sierpiński triangle #3

Square to Sierpiński triangle #4

[…]

Square to Sierpiński triangle #10

Square to Sierpiński triangle (animated)

By taking four of these square-to-Sierpiński-triangle fractals and rotating them in the right way, you can re-create the three chaos-game fractals shown above. Here’s the [v+0]-ban fractal:

[v+0]-ban fractal #1

[v+0]-ban #2

[v+0]-ban #3

[v+0]-ban #4

[v+0]-ban #5

[v+0]-ban #6

[v+0]-ban #7

[v+0]-ban #8

[v+0]-ban #9

[v+0]-ban (animated)

And here’s the [v+1]-ban fractal:

[v+1]-ban fractal #1

[v+1]-ban #2

[v+1]-ban #3

[v+1]-ban #4

[v+1]-ban #5

[v+1]-ban #6

[v+1]-ban #7

[v+1]-ban #8

[v+1]-ban #9

[v+1]-ban (animated)

And here’s the [v+2]-ban fractal:

[v+2]-ban fractal #1

[v+2]-ban #2

[v+2]-ban #3

[v+2]-ban #4

[v+2]-ban #5

[v+2]-ban #6

[v+2]-ban #7

[v+2]-ban #8

[v+2]-ban #9

[v+2]-ban (animated)

And taking a different route means that you can find more fractals — as I will demonstrate.

Previously pre-posted (please peruse):

• Square Routes
• Square Routes Revisited
• Square Routes Re-Revisited
• Square Routes Re-Re-Revisited
• Square Routes Re-Re-Re-Revisited
• Square Routes Re-Re-Re-Re-Revisited

Dilating the Delta

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

A circle with a radius of one unit has an area of exactly π units = 3.141592… units. An equilateral triangle inscribed in the unit circle has an area of 1.2990381… units, or 41.34% of the area of the unit circle.

In other words, triangles are cramped! And so it’s often difficult to see what’s going on in a triangle. Here’s one example, a fractal that starts by finding the centre of the equilateral triangle:

Triangular fractal stage #1

Next, use that central point to create three more triangles:

Triangular fractal stage #2

And then use the centres of each new triangle to create three more triangles (for a total of nine triangles):

Triangular fractal stage #3

And so on, trebling the number of triangles at each stage:

Triangular fractal stage #4

Triangular fractal stage #5

As you can see, the triangles quickly become very crowded. So do the central points when you stop drawing the triangles:

Triangular fractal stage #6

Triangular fractal stage #7

Triangular fractal stage #8

Triangular fractal stage #9

Triangular fractal stage #10

Triangular fractal stage #11

Triangular fractal stage #12

Triangular fractal stage #13

Triangular fractal (animated)

The cramping inside a triangle is why I decided to dilate the delta like this:

Triangular fractal
↓

Circular fractal from triangular fractal

Triangular fractal to circular fractal (animated)

Formation of the circular fractal (animated)

And how do you dilate the delta, or convert an equilateral triangle into a circle? You use elementary trigonometry to expand the perimeter of the triangle so that it lies on the perimeter of the unit circle. The vertices of the triangle don’t move, because they already lie on the perimeter of the circle, but every other point, p, on the perimeter of the triangles moves outward by a fixed amount, m, depending on the angle it makes with the center of the triangle.

Once you have m, you can move outward every point, p(1..i), that lies between p on the perimeter and the centre of the triangle. At least, that’s the theory between the dilation of the delta. In practice, all you need is a point, (x,y), inside the triangle. From that, you can find the angle, θ, and distance, d, from the centre, calculate m, and move (x,y) to d * m from the centre.

You can apply this technique to any fractal created in an equilateral triangle. For example, here’s the famous Sierpiński triangle in its standard form as a delta, then as a dilated delta or circle:

Sierpiński triangle
↓

Sierpiński triangle to circular Sierpiński fractal

Sierpiński triangle to circle (animated)

But why stop at triangles? You can use the same elementary trigonometry to convert any regular polygon into a circle. A square inscribed in a unit circle has an area of 2 units, or 63.66% of the area of the unit circle, so it too is cramped by comparison with the circle. Here’s a square fractal that I’ve often posted before:

Square fractal, jump = 1/2, ban on jumping towards any vertex twice in a row

It’s created by banning a randomly jumping point from moving twice in a row 1/2 of the distance towards the same vertex of the square. When you dilate the fractal, it looks like this:

square_fractal_circ_i0

Circular fractal from square fractal, j = 1/2, ban on jumping towards vertex v(i) twice in a row

Circular fractal from square (animated)

And here’s a related fractal where the randomly jumping point can’t jump towards the vertex directly clockwise from the vertex it’s previously jumped towards (so it can jump towards the same vertex twice or more):

Square fractal, j = 1/2, ban on vertex v(i+1)

When the fractal is dilated, it looks like this:

Circular fractal from square, i = 1

Circular fractal from square (animated)

In this square fractal, the randomly jumping point can’t jump towards the vertex directly opposite the vertex it’s previously jumped towards:

Square fractal, ban on vertex v(i+2)

And here is the dilated version:

Circular fractal from square, i = 2

Circular fractal from square (animated)

And there are a lot more fractals where those came from. Infinitely many, in fact.