# a qualitative one (rather than quantitative)

Paul Dirac II
Posers and Puzzles 28 Jan '15 23:14
1. 28 Jan '15 23:14
Draw a triangle.

Then randomly pick any point inside the triangle and draw a dot. Let’s call it dot A. We pick the next dot as follows. Randomly choose one of the triangle’s three corners, and draw a dot B midway between that corner and dot A. Then just iterate; that is, randomly pick another corner of the triangle and draw a point that’s midway between that corner and dot B. Call it dot C. Repeat many times. The more the better. (CADD might help for drawing this, if you have access to a program.)

What can you say about the pattern of dots that emerges from this iterative process?

(My source shows what happens in an equilateral triangle, but claims something similar would emerge in any triangle.)
2. 28 Jan '15 23:551 edit
Does the triangle corner "randomly picked" have to be different from the immediately preceding random corner? If it does I get a fractal pattern that is quite hard to describe, made out butterfly shapes that are made of butterfly shapes.
3. 29 Jan '15 00:50
Originally posted by iamatiger
Does the triangle corner "randomly picked" have to be different from the immediately preceding random corner? If it does I get a fractal pattern that is quite hard to describe, made out butterfly shapes that are made of butterfly shapes.
My source says, "randomly pick another corner of the triangle."

Confessing that I have not done the construction myself, I am not sure how literally to take the word "another." Could it mean "other than the last one utilized"? Maybe. Not sure.

I'll wait a bit before commenting further.
4. 30 Jan '15 06:53
In broad terms, iamatiger has identified the pattern.

I will see if anyone can sharpen it some more by providing a name. The dots converge (as it were) to a pattern that was named after someone back in the 1900s. Can anybody identify the name? (I am doubtful that the mathematician in question discovered the pattern through this random dots method, though I can't rule that out.)
5. 30 Jan '15 22:092 edits
a: Very sparse with some hexagons made out of tiny m's
b: Quite dense triangular fractal generated as follows:
{1} split triangle into 4 equilateral triangles, middle upside down triangle is empty except for a line around the border, for each of the other 3 triangles goto {1}.

For some reason that I do not understand b: is exactly the same as the pattern of odd/even numbers in pascals triangle:
http://math.stackexchange.com/questions/532797/discrete-math-combinatorics-homework-help

If it looks like a) Then the rule was "choose a random different point from the last one", if it looks like (b) Then the rule was "choose a random point of the three".

I drew the pictures in excel.
6. 30 Jan '15 23:42
Originally posted by iamatiger
a: Very sparse with some hexagons made out of tiny m's
b: Quite dense triangular fractal generated as follows:
{1} split triangle into 4 equilateral triangles, middle upside down triangle is empty except for a line around the border, for each of the other 3 triangles goto {1}.

For some reason that I do not understand b: is exactly t ...[text shortened]... ike (b) Then the rule was "choose a random point of the three".

I drew the pictures in excel.
(b) for the win!

My source's graphic looks just like the black & yellow image on your linked page.

The name I was looking for is found on that linked page, down near the bottom: "Sierpinski triangle."

(I have never used Excel to generate anything except for alphanumeric data within cells, so I am learning something new.)
7. 31 Jan '15 00:192 edits
I think it is because there is a triangle in the middle (of the triangle) that if a point is outside of, no line can be drawn that puts a point in the triangle. Then, because no line can start in that triangle, there is a triangle in each corner that cannot be reached, then because no lines can start in those triangles, there are other triangles than cannot be reached, and so-on.

The question (which I don't expect anyone knows the answer to) is, why is this EXACTLY the same pattern as the odd and even numbers make in pascals triangle???
8. 31 Jan '15 11:36
Here is my final chart
http://postimg.org/image/62rlbtzap/

The blue dots are the "Choose any corner" rule.
The red dots are the "Choose a different corner" rule.
9. 31 Jan '15 15:46
Originally posted by Paul Dirac II
(b) for the win!

My source's graphic looks just like the black & yellow image on your linked page.

The name I was looking for is found on that linked page, down near the bottom: "[b]Sierpinski triangle
."

(I have never used Excel to generate anything except for alphanumeric data within cells, so I am learning something new.)[/b]
I suspected it was going to be the Sierpinski triangle, but I used rule a. and got the butterflies instead. There seems to be some strange attractor behaviour going on, there. When I change it to rule b. I do also get Sierpinski.

I think you can safely extend your proof for equilateral triangles by assuming that they're all topologically identical, by the way; the generator rules skew with the triangle.

Has anyone here seen a Menger cube? Or the Sierpinski tetrahedron, which is a cross between the two?
10. 31 Jan '15 19:57
Originally posted by iamatiger
Here is my final chart
http://postimg.org/image/62rlbtzap/

The blue dots are the "Choose any corner" rule.
The red dots are the "Choose a different corner" rule.
Beautiful work! Thanks.
11. 31 Jan '15 20:00
Originally posted by Shallow Blue
Has anyone here seen a Menger cube? Or the Sierpinski tetrahedron, which is a cross between the two?
I have heard of and seen the "Menger sponge," which may be the same as the Menger cube.
12. 03 Feb '15 09:051 edit
A pentagon is not bad, moving (sqrt(5)-1)/2 towards the chosen vertex
bluish dots are the "choose a different vertex" rule
http://postimg.org/image/6gfw4lek3/
13. 03 Feb '15 19:311 edit
Originally posted by iamatiger
A pentagon is not bad, moving (sqrt(5)-1)/2 towards the chosen vertex
bluish dots are the "choose a different vertex" rule
http://postimg.org/image/6gfw4lek3/
Another beauty.

Going back to the triangle case, would it be easy to implement a non-equilateral triangle?
14. 03 Feb '15 23:28
Mobius strip & Menger sponge -----> Mobius sponge!

http://thedailyomnivore.net/2012/03/14/menger-sponge/
15. 04 Feb '15 08:332 edits
Originally posted by Paul Dirac II
Another beauty.

Going back to the triangle case, would it be easy to implement a non-equilateral triangle?
As the triangle changes the pattern distorts but stays topologically identical.

right angled triangle:
http://postimg.org/image/58ae1xg4z/

obtuse triangle :
http://postimg.org/image/lxbtxucqb/

All the small triangles in each fractal pattern are the same shape as the starting triangle.