Super #1 Fab Puzzle Poll - answer here!!!!!!!!!

[jwok]

This is both a puzzle and a poll (as that crazy title might imply).
(and for those who care, all zero of you, this is my first puzzle...sort of...even though its not a puzzle and I first saw it last semester...oh, just read on....)

It's a puzzle in the sense that I want you to try and figure out what'll happen.
A poll in the sense that, unless youve seen it, you probably wont get it and I'm curious to think what you think might occur.

(as for those who have seen it or might know a bit about the subject area behind it (oh, you know who you are), don't post anything, seriously, i want an unbiased poll. besides it should be funny to hear what people may think....)

Well, on with the puzzle:

Take your ordinary, everyday run-of-the-mill household equilateral triangle (and we all have those dastardly things left over from geometry...) like so -





well, maybe not like so, minus the eyeball and freaky numbers....

how about -



(a little mundane, really)

Now start in the dead center of it and follow these directions:

1) assign a number to each of the vertices.
2) randomly choose 1 of the 3 vertices
3) move halfway to that vertex and place a point at that halfway mark
4) from that new point randomly choose a new vertex.
5) move halfway from the new point to that vertex and place a point there.


Now repeat this process for, oh, say a couple of hundred (or thousand) or so times (I really don't expect you to do this, but try to imagine)

And now, the poll:
What do you think your collection of points will look like???

Maybe, more specifically, do you expect to see anything, at all, or just some random, chaotic mess, or even just a blob centered at the top vertex, perhaps????

I'm curious to what your answers will be. As a side note, I never expected the outcome, it was a helluva surprise...

[ctrlaltdel]

uhm... if i chose the same vertex all the time, ill end up just getting closer and closer to it, all points on one line. eventually, in infinity i will hit the vertex. but i guess i dont have so much time on my hands as to go on till infinity.

[Icarus]

If I understand what you are asking, and if I'm thinking about this the right way, I think you'll get a shrinking spiral toward the center of the triangle.

[jwok]

Maybe I should clarify this a little.


Start with your triangle, like so -



Now randomly, using a dice or numbers in a hat, not merely on some fancy whim, choose one of the three vertices. Each time, make sure its RANDOM!!!

For example, if my first 4 picks randomly chosen were
1) 2
2) 1
3) 1
4) 3

Then my triangle (so far) would like like (excuse the bad numbers, i did this quickly...)




I hope this clarifies it a little.

As for the answers I'm looking for, it could be anything - a blob looking like a giraffe, the face from turin, anything....say what you think. If you have to, try doing it ten or twenty times to see if anything looks right....

j

[jwok]

Aaaagggghhhhh!

My pictures didn't display.....

Might reference them.....


Try

1st pic
2nd pic


Also, I like your interpretation, Icarus. Stop back periodically to see how other people differ or agree with you. I'll show the results in a few days and what it really looks like.

[edit - cuz i can't make linx...]



[This message has been edited by jwok (edited 02-07-2002 03:54 AM).]

[ctrlaltdel]

well your pictures didnt load, but i tried it with a dice i got about 20+ points but no pattern whatsoever. theres a very roughly circular shape to the lef of the middle, theres a blob of points in one corner, a line of points in the other corner and nothing in the third corner...

edit: and your linx dont work either... but dont worry i know what you meant

[This message has been edited by ctrlaltdel (edited 02-07-2002 03:51 AM).]

[jwok]

fixed my links....

[mole]

I think I know this:

code:

---

   *


  ***


 *   *


*** ***

---

[mole]

(yeah, it's badly drawn, but if I made it any more detailed it would be too large)

[ZutAlors!]

I don't know the anser, but I'm gonna take a guess. And I'm gonna put that guess in invisible so as not to influence other readers of this thread. And then I'm gonna see if my guess matches mole's.

Guess: The points will form a sort of "Christmas tree thing" in the center of the equilateral triangle. This "Christmas tree thing" is defined by a circle in the center of the equilateral triangle (such that all points are outside the circle) and, on each side of the triangle, two arcs (thus, six arcs total). The arcs have one endpoint at the triangle vertex, one endpoint at the midpoint of the side, and are tangent to the interior circle. All points are contained between the interior circle and the arcs.

Well, how did I do?

Or, actually, after thinking about it (only slightly influenced by mole):Divide the triangle into four smaller equilateral triangles by connecting the midpoints of each side. The random points will cover the three triangles that contain vertices of the original large triangle?

[This message has been edited by ZutAlors! (edited 02-07-2002 08:10 AM).]

[impossibleroot]

I think I know where this is heading, but my conjecture is that there will be regions that points can never lie...triangularly shaped regions...

[This message has been edited by impossibleroot (edited 02-07-2002 08:31 AM).]

[Salty]

Sheesh, you go away for a while and everyone starts writing in invisible ink.

[Aaron]

http://uk.geocities.com/aro_hill/triangle

I drew this up in AutoCAD quickly to see, but the pattern is slow in occuring.

[ZutAlors!]

Very nice. I graphed this in Excel, using the RAND command to generate some points, and it was...different than I expected. What was also cool is that the pattern that emerges appears to be independant of starting position. You can even define a starting position outside of the triangle, and, except for the first few points, it'll do the exact same thing.

[daniel801]

no!!! the directions say that you should mark the points, but not the lines! what would that look like?

[daniel801]

guess: does it make a swastika shape?

[Coyote]

guess: It makes one of those weird fractal-like shapes where you see tinier and tinier replicas of the pattern turning up everywhere until you go crazy?

[Laramie]

From any point within the triangle, I can show a sequence that gets to any other point within the triangle. It follows that after an infinite number of trials, every such sequence will occur. Therefore, the entire triangle will be filled.

[Milt]

I guess a empty triangle inside a larger filled one.

[Coyote]

Milt, that was my initial guess too, I just wondered if maybe the three small triangles might also have smaller inaccesible triangles inside them, etc. etc.

[impossibleroot]

Coyote's right. Laramie, certainly there are *some* places that are inaccesible! I mean, barring the initial first point, what about the whole center area?

[jwok]

I absolutely love some of the guessess; the GL members' ingenuity never ceases to surprise me....


Anyway, keep guessing, the real answer will come shortly.

A coupla comments (the first is invisible so as not to influence...though the fact that its invisible probably compels most people to read it, anyways, sheesh, why i even use 'em is beyond me....)-


ZutAlors, you are absolutely correct; if you disregard the first 20 or so pointswhere you start makes absolutely no difference. In fact the traiangle being equilateral is irregardless as well...more to come when solution posted....


No the lines are not drawn but if you look at the endpoints of those lines, ya might notice something....

Finally, as said before, if you know the answer or suspect the answer due to some background knowledge, PLEASE do not post it or any clues leading towards it, i don't want any bias.

j

[mathgrant]

I've done this before, so I won't tell the answer, but if we know the point is in a certain region of the triangle, and we choose a corner, we just have to shrink the triangle to half size, and put the new triangle up next to the corner! The first point must be in the region formed by removing the center of the triangle, so let's shrink it half size and put it next to a vorner and see what we get.

[mathgrant]

Hide my post from last post thingie!

[sodasipper]

I agree with Laramie. Eventually the whole triangle will be filled.

[Laramie]

Okay. I've changed my mind. You can't get back to a small area in the center. To see this, imagine that you are basically at the midpoint of one of the sides (you can get infinitely close to it). You then choose the opposite corner and go halfway, but that point is beyond the center of the triangle. Thus, there is a small (circular?) area in the center that is empty except for the initial point.

[Bicho the Inhaler]

Aaaah...I can't resist any more. I have seen this before many times, but I'm going to post a picture anyway. Sorry if you were still thinking about it and if I'm ruining the puzzle, but it's been up for a few hours already. I took the trouble of finding a large picture that makes the pattern crystal clear. Now, without further ado, I give you:
(Here, I'll give you a healthy bit of spoiler space. Happy?)
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[This message has been edited by Bicho the Inhaler (edited 02-07-2002 10:25 PM).]

[impossibleroot]

Wow, longest spoiler-proofing ever and to no avail!

[jwok]

Aaaagggghhhhh!!!

Bicho, you ninny, don't spoil it yet. I swear I'll let you show it when the time is right. Geez, one simple request - don't ruin it - and people can't comprehend....

[mole]

It doesn't matter what the answer is, does it? I mean, your Illuminati friends can make the answer whatever they like! And who's going to complain? None of us will, since you've all brainwashed us with that first picture with the sevens and the eye and stuff! ANd you acted so innocent when you posted that...

[Aaron]

http://uk.geocities.com/aro_hill/triangle

Ok, updated for dots, not lines.
I can see the pattern emeging now, so don't look if you still want to guess.

(added extra image to show emergence)


[This message has been edited by Aaron (edited 02-08-2002 06:06 AM).]

[jwok]

cats being let out of the bag, despite the requests....ughhhh...

and whats illuminatti, mole???

i found that pick on the net as a joke, really

[impossibleroot]

It's hard to keep secrets from this lot, jwok. I enjoyed the idea, though... <grin>

[ZutAlors!]

OK, follow-on question, to which I think I know the answer, but I'm not quite sure:

When all is said and done, there are certain sections of the triangle where points will fall (call them, in sum, the "allowable" area) and certain sections where points will not fall (call them, in sum, the "unallowable" area). What percentage of the total area of the triangle does the "allowable" area encompass?

[impossibleroot]

"Notice that these areas go to 0 as n goes to infinity. This means that we have removed "all" of the area of the original triangle in constructing the Sierpinski gasket. But of course there are many points still left in the gasket. That is one reason why area is not a useful dimension for this set."

http://ecademy.agnesscott.edu/~lriddle/ifs/siertri/area.htm

[ZutAlors!]

Yeah, that's pretty much what I thought. It just struck me, all independent-like, that the "allowable" area summed to zero, which seems like an intriguing property of this shape.

[ctrlaltdel]

question....

looking at the picture with dots and a few triangles...

can we assume that ANY point on the diagram is the center point of a certain triangle which we get by the shown division method?

each triangle is touched by three smaller triangles. if we continue this division method, will we find a such triangle for each dot?