Sierpinski’s Triangle

This is a math one…but shouldn’t be too hard to understand most of it. I’ll make a comment when anything can get confusing.

First, let’s define some terms?

Triangle – No, just joking…

Fractal – Wikipedia defines them as “…typically self-similar patterns where they are ‘the same from near as from far’ [they] may be exactly the same at every scale…” Basically, if you look at them with the naked eye, it looks the exact same as if you look at it from a billion times zoom. Here are some famous fractals.

Here     and Here

Midpoint – Halfway between two points…For example, if you have a line, then the midpoint is in the middle of the line. The midpoint of Frankfort to Lexington would be Midway (I assume…not 100% on the accuracy…but I guess that’s where the name comes from).

Iteration – This is just how many times you do something. For example, when Santa checks his naughty and nice list, he preforms two iterations of checking. Doing the same  thing, so many times (we math people say n-times…but we just mean whatever number you want).

Sum – This is one I think everyone knows…just adding things up. Like the sum of 5 and 6 is 11. (5 + 6 = 11). The sum of all numbers from 1 to 5 is 15 (1 + 2 + 3 + 4 + 5 = 15)… Just adding like regular.

Any idea of Sierpinski’s Triangle? I hadn’t either until around like 9th grade. I heard about it then only because I had friends who thought they knew everything in the world because they would hit “random search” on wikipedia and read the first paragraph…But whatever. Here’s a picture of Sierpinski’s Triangle:

Sierpinski's Triangle

If you look really closely (Zoom a billion times that is), you would find that every time, you get the same picture. Basically what you do is start with a regular triangle and make a line connecting the midpoints of the sides, then connect the corners to the midpoint of the bottom line…like this:

Stage 0 is what I would call the n=0 iteration…or basically what you start with.

Stage 1 is what I would call the n=1 iteration…after you do what is described in the above paragraph.

Stage 2 is the n=2 iteration…after doing the same thing, but to all the new red triangles (notice you leave the upside down triangles alone…this is important).

So why am I telling you about this? Who cares? (“Pain and Sorrow” by Joe Bonamassa …check him out. Great blues guy.). Well, when I was young, I used to have fun counting stuff. I want you to count how many triangles there are in a Sierpinski Triangle.

No problem you say. I’ll just count them. Sure, fine by me. What if you had a triangle of iteration n=5 like I just drew a short hour ago? That’s a lot of triangles if you were to do it at home (one less than the first picture I showed you…the grey one).

Well, I have devised a formula for finding the number of triangles!!!

Now, as far as I can tell, I’m the only one to care about this or post it on the internet (note…this is after like 3 searches on google and looking only at the first couple postings…Hardly proof that I am the first). I am undoubtedly the millionth person to publish such a formula, because this is easy stuff…not the type of thing that I could figure out on my own…

But I want you to know it anyway. It’s an easy formula.

If you have an iteration of n=0 (Which I am defining as just drawing a triangle), then you have 1 triangle. (Clearly! as my modern algebra/number theory professor would say)

If you have an iteration of n=1, then you have 5 triangles that appear in front of you. (The 3 small ones, 1 upside down one, and the big-picture one).

If you have an iteration of n=2, then you have 17 triangles.

N=3 gets you 53 triangles. N=4, 161 triangles. N=5, 485 triangles. N=6, 1,457…and so on and so on, and N=10 gets you 118,097 triangles. After 100 iterations, you get a 1 with 48 numbers after that (thats 1 quindecillion for those who care…No Big Deal)

Where am I getting these numbers you ask? Just a second…for those of you who are feint of hear when it comes to mathematics, now’s a good jumping point. Go draw a Sierpinski Triangle, put it on the fridge, and be content with how mathematical it makes you look…(“When The Lady Sings The Blues” by Vince Gill….excellent guitar slinger.)

Here’s my formula for the number of triangles at n iterations…

1         n=0

1+4∑3^(i-1) (for i=1 to i=n)     n≥0

As for the proof…the only way I can think of it right now is through induction…and I’m tired of blogging and I’ve been messing with the fonts trying to get that stupid formula to look right for the past half hour completely serious…so, I’ll leave it at that since this is nothing groundbreaking nor new.

Not bad for the first math post I don’t think. What about you?