Monday, September 8, 2014

Pancake Spirograph

I’ve been working on making spirographic pancakes lately – Spirocakes.  Here’s the latest iteration of the “Pangraph,” a ring gear that fits over the frying pan and acts just like the old Spirograph toy:

It’s time-consuming to cut out the gears, so I didn’t want to just make random ones.  It was fun to work out which gears would make the most interesting patterns and be compatible with both the 35- and 36-tooth ring gears.

I even got to make use of the program I wrote 10 years ago to draw these patterns (technically called hypotrochoids and epitrochoids), to see what kinds of patterns I could expect to see with different gear sizes.

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Friday, August 8, 2014

Graphs in the sand

I noticed this “flying ring” toy making a nice sine wave as the wind rolled it along the beach this evening. 
sine wave in the sand

It reminded me of how the wind sometimes blows plants growing in sand, causing them to trace circles.

pict0046

Sunday, August 3, 2014

Free graphing calculator emulators

Smart phones are smaller and significantly more powerful that graphing calculators, so it makes sense to use them as such.  Unfortunately, not many good calculator apps exist (in my humble opinion), so I was excited to try out a graphing calculator emulator when I upgraded my Android phone recently.  Texas Instruments is trying to restrict emulator usage to those people who actually own a real graphing calculator by requiring the ROM files to be provided by the user rather than being included with the emulator app.  I figured this shouldn’t be a problem for me, having a drawer that looks like this.
Graphing calculator emulators

But transferring a calculator ROM to a phone isn’t fun and games.  I enthusiastically began following the steps I found online, but they ended up requiring an older (32-bit) operating system.  Luckily I had an old laptop on hand, and got a little further, but kept running into problems and never did get a ROM off a calculator.  I finally resorted to the most obvious solution and Googled it. 

I tried AndieGraph first, which happily pretended to be a TI-85 and TI-86 without problems.  You can even write your own programs, but there’s no way to import other programs.  True to the original, the function plotter takes its sweet time.
Graphing calculator emulators
Next I tried Graph89, which included this interesting note in the description:

Firmware updates (*.89u, 9xu, *.v2u, *.8Xu) which are normally used to restore the operating system of your calculator can also be used as a ROM image.

Sure enough, the operating systems freely available from TI (this TI-89 download, for example) can be used with this program.  You also get the capacity to import programs and apps, change the CPU speed and display color, among other settings.  The screen is a little nicer to look at than AndieGraph’s (and plotting functions is nice and fast), but it doesn’t support the 85 or 86.  I found that the arrow keys sometimes seem to register multiple presses, which is particularly annoying when writing programs.
Graphing calculator emulators

Tuesday, June 17, 2014

Newton’s rap battle limerick

Epic Rap Battles of History released a new classic yesterday, in which Newton battles Bill Nye.  The only way to improve on this would be if Weird Al played Newton – oh yeah, he did.  “To rebut,” Newton provides an equation he supposedly wrote, which is actually a silly calculus limerick:

9e6e9367e8ab8af3c79bb1033d493238

The integral sec y dy

From zero to one-sixth of pi

Is log to base e

Of the square root of three

To the 64th power of …

Luckily Neil deGrasse Tyson appears to finish the calculus limerick: “i"

Sunday, May 18, 2014

Cutting angles accurately

Building a playhouse for my kids, I had occasion to cut a forty degree angle in a piece of wibbly wobbly plastic.  You know, the corrugated kind you put on greenhouses.  I could have gotten out a protractor, marked 40º and extended it with a straightedge, but I suspect it could be off by a degree or two by the time the line extended to where I needed it to go.  Instead, I measured one side of the triangle I would be removing from the sheet, did a little triggety-trig and came up with another side, which I could mark to the nearest 1/8-inch,  which represents an accuracy of less than half a degree.  Because playhouses need to be precise.
1-IMAG2139

Wednesday, April 16, 2014

Enter the dragon

After all the fun I had making fractal snowflakes, I wanted to try out some different ideas with the “Transform Effect” in Illustrator.  Here I typed the word “Dragon,” and applied an effect that made three copies, rotated in multiples of 42º (I went for an arbitrary angle other than 60), and scaled up and shifted a little each time.
dragon fractal (1)
Applying the same effect again makes a kind of spiral, as you might imagine.

dragon fractal (2)dragon fractal (3)
But because the rotations are done with respect to the bottom-left corner (rather than the center of the spiral), a strange pattern begins to emerge: 
dragon fractal (4)
dragon fractal (5)
Here I tried it again, with a transformation that makes one copy rotated 80º about the lower left corner, and shifted right.
dragon fractal (6)
We might expect to see four “dragon”s when the transformation is applied again, but one of them falls perfectly on the first copy so we see only three.
dragon fractal (7)dragon fractal (8)
After the third iteration, the dragons have curled around enough to be farther left than the original, which moves the point of rotation.  Soon the fractal magic begins to occur.
dragon fractal (9)dragon fractal (10)
dragon fractal (11)dragon fractal (12)
dragon fractal (13)dragon fractal (14)
dragon fractal (15)dragon fractal (16)
dragon fractal (17)

Sunday, April 6, 2014

Frozen fractals all around

Watching the movie Frozen with my kids the other day, I was happy to hear the term “frozen fractals” in the acclaimed song, “Let It Go.”  It made me look closer at the snowflakes and ice shapes depicted in the movie – but I didn’t notice much in the way of fractal structure there. 

The archetypical snowflake images tend to exhibit “branches on branches,” indicating fractal self-similarity, though the six-fold symmetry appears to be the defining characteristic of iconic snowflake shapes. 

01f

I wondered about making my own fractal snowflakes using a simple iterative process…but I wanted something different than the Koch snowflake.
06-round1
Starting with a random shape, I repeated the following process (in Adobe Illustrator, which made it easy):

Create 5 more copies, rotated in multiples of 60º about the lower left corner of the bounding box.

Here are the first four iterations:
07-round2 08-round309-round4 10-round5
After that my computer was slowing down, but you get the idea.  I tried the same idea again, but this time the copies were rotated about one vertex so they overlapped and ended up making a cool tile pattern:

12-pointy1 13-pointy214-pointy3 15-pointy4