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## Campaign for Real Pi Day

Yes, it’s that time of the year again, the time for mathematically inclined curmudgeons to gripe about the inadequacies of March 14 as a date for Pi Day.  I’ve expounded on this in the past.  To give the proposal of an alternate date some permanence, I’ve created a site to promote Real Pi Day.  This year, the Chrono-Solsticial Pi Point will be at Saturday, April 17, 2010 at 12:02 AM Coordinated Universal Time.

Visit the website at www.realpiday.org for more details.

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## Nuts to Pi Day

Well, it seems to be time for another simpering genuflection to the Gregorian calendar in the guise of lauding a transcendental constant. This is the same thinking that causes the doomsday chiliasts (to use Carl Sagan’s phrase) to get all het up any time there’s a date with three or more zeroes or sixes.

I’ve railed against the Vulgar Pi Day before, so I shan’t re-rant.

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## Happy Chrono-Solsticial Pi Day

Today, is, of course, the day when we have progressed through one pi-th of the time between the last winter solstice and the next. Vulgar Pi Day is observed by plebes and math groupies on, depressingly, 3/14. As has been discussed previously in this forum, the true, transcendal observance requires a bit more thought.

Writing these few sentences has sorely taxed my depleted mental resources. What little remains of my sanity must be preserved for the final qualifying exam I shall be taking tomorrow afternoon. Soon, this ordeal will be over, to be replaced a completely new and more complicated ordeal. Huzzah!

(Image of the sun on the winter solstice, 2006 C.E. courtesy of geo3pea.)

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[Update, April 13, 2007: Thanks to Herr Ziffer for catching a confusing typographical error.]

I can’t believe I’d never seen (or figured out) this quick method for converting a binary number to a decimal number in your head. All you need to be able to do is double numbers and occasionally add one.

1. Start at the first ‘1’ on the left, and start with the mental number one
2. Move one digit right. If that digit is a zero, multiply your mental number by two. If it is a one, multiply your mental number by two and add one.
3. Repeat step 2 for every digit of the binary number

Here’s an example. We’ll use the binary number 1101010 1011010:

• 1011010 – We start at the first one. Our mental total: 1
• 1011010 – Next digit is a zero; we double our mental number: 1 x 2 = 2.
• 1011010 – Next digit is a one; we double our mental number and add one: 2 x 2 + 1 = 5
• 1011010 – Another one; double and add one: 5 x 2 + 1 = 11
• 1011010 – Zero; double: 11 x 2 = 22
• 1011010 – One; double and add one: 22 x 2 + 1 = 45
• 1011010 – And finally a zero; double: 45 x 2 = 90

The rest of this post is a little more technical, so if you glazed over when reading the above, it now may be time to soothe your tired mind.

I happened across this trick while contemplating a three-state discrete finite automaton that identifies binary numbers divisible by three. The automaton starts in state 0, and like the above procedure starts at the left side of the number. The number of the state can be thought of as the remainder of the number as read so far, mod 3. Every time a zero or a one is read, the automaton follows the arrow with that label from its current state. If it ends in state 0, the number is evenly divisible by three. Once I understood why the DFA actually works, the mental calculation became glaringly obvious.

For even more fun, the regular expression (0*(1(01*0)*1)*)* will also match binary numbers divisible by three.

Exciting! Now you have something to talk about the next time you go to a cocktail party.

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## Mental Exercises: Multitasking with Numbers

The following exercises build on those described in the post on mental exercises with number sequences.

• Visualize a number sequence: select any of the sequences from the previous post, but rather than simply “counting” or saying the number aloud, form an image of each element on the sequence in your mind.
• Count one series aloud or silently while visualizing a different sequence.
• Count one series aloud or silently while writing a different sequence.  (Any or all of these can be single or multiple sequences.)
• Recite a series as in the previous exercises, but in a different base: count by 5’s in octal, by 3’s in base 11, by 7’s in hexadecimal.
• Visualize a scene from your life (such as a walking downstairs or through your neighborhood, or going to a restaurant, etc.) while reciting a number sequence.

This is the majority of the useful exercises from the relevant section of the Wujec book.  Once my copy of Orage arrives, I will be quite interested to see how the 200 exercises contained therein are presented.

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## Tetrahedra

I’ve been rereading R. Buckminster Fuller’s Synergetics thanks to a Christmas gift. Thus primed, I was amused to come across two tetrahedron-related posts on BoingBoing:

Bucky was, of course, fascinated by tetrahedra:

113.00 When we take two triangles and add one to the other to make the tetrahedron, we find that one plus one equals four. This is not just a geometrical trick; it is really the same principle that chemistry is using inasmuch as the tetrahedra represent the way that atoms cohere. Thus we discover synergy to be operative in a very important way in chemistry and in all the composition of the Universe. Universe as a whole is behaving in a way that is completely unpredicted by the behavior of any of its parts. Synergy reveals a grand strategy of dealing with the whole instead of the tactics of our conventional educational system, which starts with parts and elements, adding them together locally without really understanding the whole.

(From the online text of Synergetics.)

For more tetrahedral goodness, check out these instructions for making a model of five interlocking tetrahedra from Thomas Hull. I’ve made several of these, and they’re great fun (on the most recent, the tolerances were a bit off, so it ended up being four interlocking tetrahedra). Here’s a nice image of a completed model.

If you’re not up to modular origami, you can also try this printable PDF papercraft version of five tetrahedra as a compound solid.