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.

1-Fullscreen capture 982014 105959 AM

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

Wednesday, March 26, 2014

Happy birthday, Paul Erdős!

Here’s a caricature I made of “the man who loved only numbers” for his 101st birthday. 
Paul Erdos caricature

Wednesday, March 12, 2014

Pi Day Challenge

Here’s a fun site for your lateral thinking students (who also know some geometry).  In their words,

A team of logicians adapted or created these puzzles - some require research, some require mathematics, some require pure savvy.

Pi Day Challenge

pi day challenge

A lot of the problems don’t have anything to do with pi, and some of the presentation is a little janky, but it’s still fun.

Tuesday, March 4, 2014

Pi art ideas

I recently came across some cool ideas for blending up art and math for younger kids at whatdowedoallday.com – just in time for Pi Day next Friday!
math-art-pi-starting

Sunday, March 2, 2014

Permutation of the Bells

I came across this picture I took while visiting the bell tower of the Old Post Office in Washington, D.C. a couple of years ago.  It describes how the bells are occasionally rung in a “full peal,” consisting of all possible permutations of their tones.  With 6 bells, you get 6! = 6*5*4*3*2*1 = 720 permutations, which could be rung in under an hour, but with 7 bells, you get 7! = 7*720 = 5040 patterns to ring!

I thought maybe this would involve listing the set of permutations in increasing numerical order:

123456
123465
123546
123564
123645…

But In the example they showed, it seems that the pattern is to first swap neighbors in 3 groups:

12 34 56
21 43 65

And then to keep the first and last positions in place but swap neighbors in the 2 groups inside:

2 14 36 5
2 41 63 5

--

I can’t tell if that’s all there is to the method or not, and I wonder if there’s a proof that the method produces all permutations.  A quick test with 4 bells shows that you don’t get all 24 permutations before it repeats.

1234
2143
2413
4231
4321
3412
3142
1324
1234

Saturday, February 22, 2014

Math murals

Last weekend I had the opportunity to help my old math department finish up the digits of pi mural we began in 2009. 

pi digitspi mural
While I was there I convinced them to let me put Archimedes on the wall by the staff bathrooms (even though we don’t have bathtubs in them).
archimedes mural

Friday, January 24, 2014

System of 3 equations APPLIED

A lot of times the applications we come across for algebraic topics feel contrived; here’s one that occurred naturally in my very own home yesterday. 

My wife works in a clinic with two other midwives.  They all do the same amount of “on-call” time, but have different workloads when it comes to clinic hours. And sometimes they’re on call while they’re in the clinic.  She knows their salaries, but didn’t know the hospital’s pay rate for clinic hours, on-call hours, or combo hours – she was interested in finding this out to see if it was worthwhile to pick up some extra shifts at the clinic.

system of 3 equations

Calling the number of on-call hours X, the number of clinic hours Y, and the number of combo hours Z, she set up an equation for each midwife and was able to figure out the pay rate for each type of work. 

Math to the rescue!

Wednesday, January 1, 2014

Doomsday 2014

Well, we've evaded the apocalypse for another year.  2014 has its own Doomsday to keep in mind, however - and it's a Friday.



Dr. Conway developed a sweet method (called the Doomsday Algorithm) to determine the day of the week for any day in history.  It takes some practice, but dealing with the current year is pretty simple, provided you know the Doomsday.
Doomsday 2013 is Friday*.
  • This means that the following dates are all Thursdays: 4/4, 6/6, 8/8, 10/10, 12/12
  • If you work nine to five at the local 7-11, you can remember four more Thursdays: 9/5, 7/11, 5/9, 11/7.
  • Einstein’s birthday = Pi Day = 3/14 = Thursday.
  • The last day of February (2/29 this year) is a Thursday.
  • Lastly, 1/3 is Thursday (3 out of 4 years it’s the 3rd… but on leap years it’s the 4th).
What day of the week is your birthday?  Independence Day?  Halloween?  Christmas?

Adding days of the week is easier when you treat them as numbers: mONEday, TWOsday, 3ednesday, FOURsday, 5day, S6turday (Treat Sunday as 0 - "nunday").  What's ten days past Wednesday?  10 + 3 = 13.  Take out any multiples of 7 and you have 6 (Saturday). 

*In case you’re interested, the link up top does a thorough job of teaching the method, but here’s the quick version. Start with the century day.  For 2000 it is Tuesday (2).  Then look at the 2-digit year, 14, and ask how many 12’s are in it. (1). What is the remainder? (2) How many 4’s are in this number? (0). Add up (2)+(1)+(2)+(0) = 5 = 5th day of the week = Friday. 
Craziness? How about one more example from the future: 2055.  Century day is still (2).  How many 12’s in 55? (4). What is the remainder? (7).  How many 4’s in this number? (1).  Add ‘em up: (2)+(4)+(7)+(1) = 14.  Take out multiples of 7 and you get 0.  The 0th day of the week is Sunday. Now you can find the day of the week for any date in 2055.