Pastelland Forum

Good luck.

https://www.youtube.com/watch?v=ZKvEFZbmp7M
Now that's what I call murderous maths.

Ah, the math thread.

Here's my Google Drive of Pisano stuff (updated as I complete things). Discuss!

https://drive.google.com/open?id=0B11R3 ... npNS1hFcEE

It looks very complete why the names Pi and Tau?

By the way, if there is any mathematical question you have, or you noticed some pattern and want to know how to prove it, this website can be of help: https://math.stackexchange.com/
I've asked some questions there myself and always receive great answers.

Vortex wrote:It looks very complete why the names Pi and Tau?

By the way, if there is any mathematical question you have, or you noticed some pattern and want to know how to prove it, this website can be of help: https://math.stackexchange.com/
I've asked some questions there myself and always receive great answers.

Pi - The first exploration I did on Pisano symbols (the original symbols)

Tau (2Pi) - The second set of symbols generated through the second exploration I did,

3Pi - the third thing I did with the midpoints.

Really I just wanted to find a temporary way to group the three different symbol types so I slapped some terms down.

Ok, so after having secret conversations of secrecy with secret people about more Fibonacci stuff I came across this paper someone wrote for his Master's thesis talking all about the Fibs and mods related to them:

He has a general page with all these links on it here: http://webspace.ship.edu/msrenault/fibonacci/fib.htm

Thesis (interesting part concerning the zeros of a period is Section 3.3): http://webspace.ship.edu/msrenault/fibo ... Thesis.pdf

He also introduced a new concept, at least to me, in Section 4.1 called "spirolaterals" and tried to apply them to the Pisano periods, but I don't know how cool his results were because the pictures don't appear to actually be in the document.

More on Spirolaterals: http://thewessens.net/ClassroomApps/Mai ... etry&id=10

However, what was more interesting was that he found a way to 'plot' the pisano periods as a series of points where you take the numbers in the sequence and pair them up so you have (1st term, 2nd term), (2nd term, 3rd term), and so on, and you get grids of points that make cool patterns. Link: http://webspace.ship.edu/msrenault/fibo ... od%20m.pdf

I personally tried it out with my bb #10 and got:
The interesting thing about 10, at least, is that you could technically "repeat" this pattern by extending the axes and "modding the axes" by (mod 10) as well, and just keep copying the pattern in all directions, and it will make a seamless tesselation. I also connected some of the lines that I thought were more obvious and it sort of develops into a nice quilt pattern.
I'll probably do more graphs in the future when I have spare time just to see how they look.

that looks pretty cool

The interesting thing about 10, at least, is that you could technically "repeat" this pattern by extending the axes and "modding the axes" by (mod 10) as well, and just keep copying the pattern in all directions, and it will make a seamless tesselation.

wouldn't that happen with all of them, not just 10? or am I misunderstanding what you say?

Apocrypha wrote: More on Spirolaterals: http://thewessens.net/ClassroomApps/Mai ... etry&id=10

403 forbidden error

Empoeria wrote:

Apocrypha wrote: More on Spirolaterals: http://thewessens.net/ClassroomApps/Mai ... etry&id=10

403 forbidden error

works fine for me.

Vortex wrote: wouldn't that happen with all of them, not just 10? or am I misunderstanding what you say?

No; if you look at the last link you'll see that some examples given especially later on in the document might have rotational symmetry but if you were to copy them along the axes in any direction there's no tesselation pattern that continues between them. I'm looking at ones like m=39 and the rotational ones like m = 82, 221, 257.