- Your quiz returned.
I'll bet that you're feeling like I am: thank Ra I wasn't born an Egyptian!
I'll not antagonize you with Egyptian division any more; but you'll have to do the multiplication again....
- The quiz this week will be over Babylonian and Mayan representations of numbers.
You should able to write our numbers in either system, and translate their numbers into our own.
- Next week, Thursday, we have our first exam (on Thursday, 2/27). Start studying up! Make sure that you do your readings, as they will be a part of the exam.
- We did a few more Babylonian numbers
- 10,000 as a Babylonian number?
- How about 216,000?
Did you ever notice how we block numbers into threes?:)
- We continued with the Mayan culture: and we'll wrap that up right now!
How did the Mayans represent numbers? - Based on this artifact, the Xultun (shool-TOON) lunar calendar:
- We have deduced that
- A dot represents 1
- A bar represents 5
- It seems like it's a base 20 system, but the third place might be a "360" place (we were expecting \(20^2=400\), if it were an honest base-20 ("vigesimal") system).
- Sean noticed that the first four numbers at the top were just multiples of 177. But that eventually fails. In which column?
- What is that other symbol?
- If you recall from Zero, My Hero, there are
people who "counted on their fingers and toes...."
A base 20 system is called "vigesimal". The Mayans had
a quirk in their system, however, which rendered it an
impure base-20 system (you'll know this, if you did
your reading!). It's a "mostly vigesimal" system. But
the third place should have been \(20^2\): what is it?
(Column c can tell us....)
After that, however, the Mayans used multiples of 20 to
increase the numbers in the next positions: 1, 20, 360, 360*20=7200, 360*202=144000, etc.
- How would the Mayans write these numbers:
- 47
- 400
- 573
- 3600
- 7200
- 14001
- "All that remains now is for us to discuss the reasoning
behind the 177- and 178-day intervals of the reconstructed
Xultun lunar table (Figure 11).
It has long been known that pages 51-58 of the Dresden Codex contain a lengthy lunar table, sometimes called the Dresden Codex eclipse table (Figure 12). The Dresden Codex calculates moons in intervals of six lunations (177 days), with an occasional correction of five lunations (148 days) over a period spanning 11,958 days, or 405 total lunations (see Aveni 2001:173-184 for a thorough discussion). This works out to an average of 29.52593 days per lunation [ael: about 42517 minutes]. Mayanists have long marveled at the accuracy of this table, which places the average length of a lunation within seven minutes of the modern figure of ~29.53059 (Aveni 2001:183). More fascinating still, however, is that the recently discovered Xultun table actually comes even closer to the modern value. It will be remembered that the Xultun table covers a total of 4,784 days and represents 162 lunations (Saturno et al. 2012:715), which works out to a mean length of 29.53086 days per lunation, or within four minutes of the modern figure."
(from Unearthing the Heavens: Classic Maya Murals and Astronomical Tables at Xultun, Guatemala)
The Mayans were very good at counting!
- We have deduced that
- Based on this artifact, the Xultun (shool-TOON) lunar calendar:
Today we move on to the game I will call "Fraudini Nim".
I've told you that I like to give you ways to make money, and this is the best one I've got. So learn this game! Play it well, and you'll make a lot of money over your lifetime. You'll also score points on your next exam....
- Rules:
- There are two players, Player 1 and Player 2.
- An arbitrary number of counters is thrown down (e.g. toothpicks, pennies).
- Objective: The player who picks up the final counter wins.
- Player 1 will go first. On the first play, Player 1 must pick up at least one counter, but may not pick up all the counters (otherwise this would be a silly game!).
- Player 2 must then pick up at least one counter, and not more than twice the number of counters picked up on Player 1's turn.
- The players then alternate turns, subject to these same conditions: pick up at least one, no more than twice what the preceding player took.
- The player who legally picks up the final counter on their turn wins.
- So I'm going to play a game against someone, who can win my
money. You don't have to put anything down, because I'm going
to beat you anyway. Who wants to play, and thinks maybe they
can win my money?
- Okay, so what's the strategy? How does one win at Nim? How could I
be so confident that I was going to beat anyone?
- Let's try to figure out the strategy, by creeping up on it. What
do I mean? Start with some really simple games....
- If we look at some of the simplest cases, what can we
conclude?
Number of counters 1 2 3 4 5 6 7 8 Winner (1 or 2? -- if they play right!) X - What's the winning strategy in Fraudini Nim?
- If we look at some of the simplest cases, what can we
conclude?
- A review of the Mayan calendar we study, Unearthing the Heavens: Classic Maya Murals and Astronomical Tables at Xultun, Guatemala , including the reference to this article from the journal Science: Ancient Maya Astronomical Tables from Xultun, Guatemala, and this more "popular" interpretation of the calendar.
- Do math like an Egyptian! (Walk Like An Egyptian/Michael Jackson Mash-up!)
- Hexagonal Paper
- Pascal's triangle (wikipedia)
- American Bandstand dances to Gimme Some Lovin', a hit by The Spencer Davis Group: here's their version.
- List of musical works in unusual time signatures
- Blue Rondo a
la Turk, by Dave Brubeck: illustrating
how to play a piece in 9/8.
Watch Dave Brubeck's leg, counting out the rhythm....