$1.98 Mathematics, Part 2

A couple years ago, I was walking about a 99-cent store and found a quad-ruled composition notebook and a box of colored pencils. Total: $1.98. I played around with them for a while, and then they were put in a draw and forgotten about. Until recently when I found the notebook. And then last week, I posted a sketch for those pages.

Here is another one:

This was actually the first sketch from the book, but is wasn't as colorful as the other one. And it seemed to be more boring. But is it really?

The sketch (and you can click on the image for a larger version) shows a Golden Spiral created by connecting the diagonals of adjoining squares. The length of the sides of each square are determined by using the next number in the Fibonacci sequence. Obviously, the squares increased in size so quickly that I couldn't finish the 34 x 34 square.

But there was something else I noticed. I had added extra diagonals to some of the rectangles that were created in addition to the squares. I highlighted one of them in red (on the scan -- it's in pencil on the original sketch). The red line appears to be the diagonal for many of the rectangles. Four of them, in fact.

How could that possibly be the case?
(If any of my students are reading this, STOP here and look at it. Investigate. See if you can figure it out. Come back when you have it or you've had enough. I'll wait.)

The four rectangles have the following sizes: 2 x 1, 5 x 3, 13 x 8 and 34 x 21. Zooming in shows that the red line really isn't a diagonal of the smallest rectangle, so let's discard that one for a moment. The others are close enough to be errors in sketching. Since the slope of a straight line has to be constant, if we calculate the slope at any two points, we should get the same number.

Slope can be calculated as rise over run or change in vertical over change in horizontal. (You remember that "delta y / delta x" thing I keep mentioning in class? Yeah, that.)

So we have slopes of 3/5, 8/13, and 21/34, which are definitely not equivalent fractions. (How do we know that?)

If we convert those fractions to decimals, look what we get:3/5 = 0.6
8/13 = 0.615384...
21/34 = 0.617647...
and if the paper had been bigger, we might have seen
55/89 = 0.6179775...
144/233 = 0.61802565...

So the slopes are nearly identical meaning that the diagonal of the big rectangle isn't really the diagonal of the others, but it's really, really close.

Extra points if anyone keeps going, or if they can tell me the significance of a particular number that starts 0.61803...

Top of the Top 50 Mathematics Blogs!

I found out yesterday that (x, why?) is once again on the Top 50 Mathematics Blogs list published by the Guide to Online Schools.
I found out this morning that it was the top of the list! Number One on their Favorite Five:

Our Favorite Five (x, why?): Christopher Burke posts math-related comics, jokes, and general discussion. Why we love it: Burke's comics and discussions show the lighter side of formulas and curves.  Favorite Post: Benoit Mandelbrot

Edit: Link added.

Almost Right -- Spiked Version

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


I don't usually do alternate versions of the same strip, but once I thought of this, I couldn't not do it.

Spiked Math

Almost Right

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


If Bob is a isosceles triangle and his legs have a length of two, then you could call him Bob Twos Triangle.

$1.98 Mathematics

A couple years ago, I was walking about a 99-cent store and found a quad-ruled composition notebook and a box of colored pencils. Total: $1.98. I played around with them for a while and then they were put in a draw and forgotten about. Until recently when I found the notebook.

Here's one of the pictures:

There's a few things going on here. First, there's the demonstrations that two triangular numbers make a square. Second, the overall illustration shows that the sum of consecutive cubes is equal to the square of a triangular number.

Written on paper, there seems to be no reason why:
1³ + 2³ + 3³ + 4³ + 5³ = 15²,
(with 15 being the 5th triangular number)
But the visual shows it to be true. There is one box in the corner, bordering two 2x2 boxes, bordering three 3 x 3 boxes. Granted, I cheated in that all the even numbers contain two rectangles that are 1/2n x n.

I didn't finish coloring it. Probably got bored. Likewise, in the actual notebook, the bottom of the page has four 9 x 9 boxes lightly penciled, but the other five would be off the edge of the paper.

Now that I found the book again, I may start doodling some more... assuming I find the colored pencils.

EDIT: I replaced the image with an annotated version. The original, larger image (click on the picture) is still annotation-free.

Refreshments

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


Oddly, the size of the popcorn is a constant and independent of everything except the realization that I don't know why I'm still eating it.

Medium

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


Another 100 lbs. and it'll be the Extra Large Medium's Mall. Say it out loud.

"Peebles Lab" Turns 100

Andrew Stella from Peebles Lab webcomic is celebrating its 100th comic. As a tribute, he did some guest art for some other comics, including the one below:




Those two guys look familiar, but I just can't place them.
Now, that's one pun I'll have to scratch off my "To Do" list.

Geometrica: The Second Dimension

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


This will cause complications for future productions as the Ophiuchians demand their rightful seat at the Council.

Check out the previous appearance of Geometrica.

Mickey Mouse's Triangle

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


Why was I investigating Sierpinski triangles and how did I happen to notice something that looked like a "hidden Mickey"?
Let's just say, "No reason in particular" because that'll probably be the least weird answer.

Basically, the exact numbers in Pascal's Triangle aren't important. You only need to know if they are divisible by 3, or if they have a remainder of 1 or 2. You can assign color codes to them, and then have a handy reference chart for adding.

I'll give it two hours tops before someone finds a mistake like adding yellow and red and getting another yellow instead of black.

Cylinder

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


Sometimes you just have to turn to the world around you for inspiration.
Just don't be "inspired" when it's time for me to make coffee!

Recreational Mathematics: Knots and Stars

I did a little investigating of both stars and knots for comic that should have been made back in the summer of 2009, but nothing came of either ... yet.

I'm reminded of this because of two videos on Vi hart's blog. Namely:

• Doodling Snakes and Grpahs, and
• Doodling Stars

.

Both are fun to watch even if you don't try them out. You might even want to go through them a couple of times. You never know if you missed anything!

i Before e

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


Or when calculating Euler's Identity.

Calling Names

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


Should've bopped him right between the square root of negative four!

Happy New Year 1/1/11

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


Better. Stronger. Faster.

And he has a good reason, but we'll have to save that for another comic.