Polyhex — An analog of the polyominoes and polyiamonds in which collections of regular hexagons are arranged with adjacent sides. They are also called hexes, hexas, or polyfrobs.
For the 4-hexes (tetrahexes), the possible arrangements are known as the bee, bar, pistol, propeller, worm, arch, and wave.
Vampires can’t exist. Why? Because they’d quickly depopulate the earth.
To prove it, the scientists do some calculations by picking a random year in history — 1600, specifically — and imagining what would happen if one vampire suddenly appeared on earth. They assume, for the sake of argument, that a vampire needs to feed “only once a month”, and that in the course of feeding, the vampire turns its victim into another vampire. They note that the global population of humans was 536,870,911 in the year 1600.
Then the calculations begin. If a single vampire fed on a single human in the first month, this would create two vampires — and decrease the human population by one, leaving it at 536,870,911 - 1 = 536,870,910. In the second month, those two vampires would each feed, transforming two people into vampires — so you get four vampires and a human population of 536,870,911 - 3 = 536,870,908. So you can see where this is headed. The vampire population is increasing in a geometric progression, and the population of humans is similarly decreasing — and at that rate, the authors calculate, the entire human population would be transformed into vampires in only 30 months. QED!
Sure, humans could increase their numbers by having children — but the birth rate could never keep pace.
We conclude that vampires cannot exist, since their existence contradicts the existence of human beings. Incidently, thelogical proof that we just presented is of a type known as “reductio ad absurdum”, that is, reduction to the absurd.
{ collision detection | Continue reading }
I don’t trust scientists and I don’t trust mathematical formulas, and therefore I will continue to wear a string of garlic around my neck while carrying a 2 foot wooden stake where ever I go. This is obviously a vampire-funded study that aims to lower our guards.
Think too hard about it, and mathematics starts to seem like a mighty queer business. For example, are new mathematical truths discovered or invented? Seems like a simple enough question, but for millennia, it has provided fodder for arguments among mathematicians and philosophers. Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?
On the other hand, if math is invented, then why can’t a mathematician legitimately invent that 2 + 2 = 5? (…)
Plato is the standard-bearer for the believers in discovery. The Platonic notion is that mathematics is the imperturbable structure that underlies the very architecture of the universe. By following the internal logic of mathematics, a mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. (…)
If the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it. (…) Brian Davies, a mathematician at King’s College London, writes that Platonism “has more in common with mystical religions than with modern science.” And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong. He titled his article “Let Platonism Die.”Davies argues that brain-imaging studies are making this belief steadily less plausible. He points out that our brains integrate many different aspects of visual perception with memory and preconceptions to create a single image — not always correctly, as optical illusions make clear. He also says that brain-imaging studies are beginning to show the biological basis of our numeric sense.
illustration { Damien Hirst, The Holy Trinity, 2005 }
| The invention of printing and the first appearance of the equal sign (=)
Johann Gutenberg’s invention of the printing press (around 1440) revolutionised mathematics, enabling classic mathematical works to be widely available for the first time. Previously, scholarly works, such as the classical texts of Euclid, Archimedes and Apollonius had been available only in manuscript form, but the printed versions made these works much more widely available. At first the new books were printed in Latin or Greek for the scholar, and many scholarly editions appeared. (…) The invention of printing also led to the gradual standardisation of mathematical notation. In particular, the arithmetical symbols + and – first appeared in a 1489 arithmetic text by Johann Widmann. Surprisingly, the symbols × and ÷ were not in general use until the seventeenth century. Robert Record, probably the most important writer of textbooks in English (…) introduced several entertaining terminologies that didn’t catch on, such as sharp and blunt corners for acute and obtuse angles, touch line for a tangent, and threelike for an equilateral triangle, but he also introduced the term straight line, which is still used. Record’s most celebrated piece of notation made its first appearance in the Whetstone of witte of 1557. Here we find the first appearance of our equals sign: And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorde use, a parre of paralleles, o: Gemowe lines of one lengthe, thus: == because noe 2 thynges can be moare equalle. |
Renaissance art and the rise of geometry
One notable feature of Renaissance painting was that, seemingly for the first time, painters became interested in depicting three-dimensional objects realistically, giving visual depth to their works, as contrasted with earlier works such as the Bayeux tapestry where such depth is not to be found. This soon led to the formal study of geometrical perspective. The first person to investigate perspective seriously was the artisan-engineer Filippo Brunelleschi, who had designed the self-supporting octagonal cupola of the cathedral in Florence. Brunelleschi’s ideas were developed by his friend Leon Battista Alberti, who presented mathematical rules for correct perspective painting and stated in his Della pittura [On painting] that ‘the first duty of a painter is to know geometry’. Piero della Francesca was another who investigated mathematical perspective. In particular, he used a perspective grid in his investigations into solid geometry, and wrote books on the perspective of painting and the five regular solids. This 1472 picture, his Madonna and child with saints, shows his mastery of perspective. Another work of the time was a 1509 book On divine proportion on regular polygons and polyhedra by Piero’s friend Luca Pacioli, whom we’ll meet again later. The woodcuts of polyhedra for this book were prepared by Pacioli’s student Leonardo da Vinci, who explored perspective more deeply than any other Renaissance painter, and whose notebooks contain much of mathematical interest. In his treatise on painting, da Vinci warns ‘Let no one who is not a mathematician read my work’. Albrecht Dürer was a celebrated German artist and engraver who learned perspective from the Italians and introduced it to Germany. He produced a number of drawings showing how to realise perspective, and his famous engravings, such as St Jerome in his study, show his effective use of it. His Melencolia is also well known, and features a number of mathematical items, such as a truncated tetrahedron and a 4 × 4 magic square in which the date of the engraving (1514) appears in the middle of the bottom row. |
{ Professor Robin Wilson | Continue reading }
artwork { Piero della Francesca, Diptych of Battista Sforza and Federico da Montefeltro, 1472 }
new york craigslist
Celebrity Nail clippings for Elementary Math tutor
Date: 2007-05-01, 12:31AM EDTI work at a very prestigious nail salon in new york, with an a-list clientel. I have a collection of nail clippings from various clients such as Cameron Diaz, Gwen Stefani, Beyonce and Scarlett Johansson. My son who is in 7th grade is in desperate need of a Math tutor. I live in Manhattan and I would be willing to meet at a mutual location with my son. I will be willing to trade my collection for four one hour sessions. Serious inquiries only please. Thank you.
The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids: the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, as was proved by Euclid in the last proposition of the Elements.
related { Are our brains wired for math? }
You have a pile of 24 plates. Twenty-three of them have the same weight, and one is heavier. You must figure out which single plate is heavier and do it with the minimum number weighings. You have a balance scale that compares the weight of any two sets of plates out of the total set of 24 plates.
How many weighings are required to identify the heavier plate?
[solution in the Comments section]
photo { Bill Owens }
When I was a kid and my dad would take us on road trips up to Karuizawa, my brother and I played the license plate number game. The rules are simple: Using basic math, make the four digits on the license plate equal 10.
In this case, you could do:
1+2+3+4=10
or
1×2x3+4=10{ Tokyo Mango }
A Spanish physicist has discovered what appears to be a new law of nature that may help explain, among other things, how life arose out of a primordial soup and why investing in losing stocks can sometimes lead to greater capital gains.
Called Parrando’s paradox, the law states that two games guaranteed to make a player lose all his money can generate a winning streak if played alternately.
Named after its discoverer, Dr. Juan Parrando, who teaches physics at the Complutense University in Madrid, the discovered paradox is inspired by the mechanical properties of ratchets — the familiar saw-tooth tools used in automobile jacks and in self-winding wristwatches. (…)
The paradox is illustrated by two games played with coins weighted on one side so that they will not fall by chance to heads or tails.
In game A, a player tosses a single loaded coin and bets on each throw. The probability of winning is less than half.
In game B, there are two coins and the rules are more complicated. The player tosses either Coin 1, loaded to lose almost all the time or Coin 2 loaded to win more than half the time. He plays Coin 1 if his money is a multiple of a particular whole number, like three.
If his money cannot be divided evenly by that number, he plays Coin 2. In this setup, the second coin will be played more often than the first.
‘’Sure enough,'’ Dr. Abbott said, when a person plays either game A or game B 100 times, all money taken to the gambling table is lost. But when the games are alternated — playing A twice and B twice for 100 times — money is not lost. It accumulates into big winnings. Even more surprising, he said, when game A and B are played randomly, with no order in the alternating sequence, winnings also go up and up.
Because these results seem so surprising, the outcome is paradoxical — Parrando’s paradox. Switching between the two games seemed to create a ratchet-like effect. With its saw-tooth shape, a ratchet allows movement in one direction and blocks it in the other.
