Monday, March 25, 2013

Automata on the beach

I found this shell on the beach in Saipan last year.

Its pattern reminded me of the some of the cellular automata I had seen in Wolfram’s A New Kind of Science (chapter 3) – it turns out that this is no coincidence.  The process by which some species of mollusks color their shells can be modeled by cellular automata: as the shell grows, the coloring of each row is affected by the coloring of the previous row through a small set of rules.

Dr. Coombes of the University of Nottingham gives a brief overview of some of the mathematics behind the structure and pigmentation of seashells here.

Wednesday, March 6, 2013

Here’s a fun idea to talk about with your students this Pi Day:

What else can you do on Pi Day? As always, www.piday.org has some good ideas for you.  I like to have students do Bouffon’s needle experiment with toothpicks and compile the results with multiple classes.

Discuss why an accurate value might be necessary, and some historical approximations:
•3.16045 (=16²/9²) Egypt, 2000 BC
•3.1418 (average of 3+1/7 and 3+10/71) Archimedes, 250 BC
•3.125 (=25/8) Vitruvius, 20 BC
•3.1622 (=√10) Chang Hong, 130 BC
•3.14166 Ptolemy, 150
•3.14159 Liu Hui, 263
•3.141592920 (=355/113) Zu Chongzhi, 480
•3.1416 (=62832/2000) Aryabhata, 499
•3.1622 (=√10) Brahmagupta, 640
•3.1416 Al-Khwarizmi, 800
•3.141818 Fibonacci, 1220
•3.14159265358979 Al-Kashi, 1430

A digit contest at the end of the day is always fun too.  Here’s the hand-out I give to kids a few days in advance if they want to try memorizing for fun (it has a spacer every 50 digits):