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June 25, 2006
~Putting all reason aside you exchange / What you got for a thing that's hypnotic and strange~
Number spirals, with many pretty diagrams and mind-blowing visual math:
Number spirals are very simple. To make one, we just write the non-negative integers on a ribbon and roll it up with zero at the center.
The trick is to arrange the spiral so all the perfect squares (1, 4, 9, 16, etc.) line up in a row on the right side.
It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.)
It turns out that every integer on a product curve is intersected by a second related curve.
The existence of the second curve is a consequence of the following property of offset curves in general: the product of any two adjacent integers on a curve can be found further out on that curve at a distance from the first factor which is equal to the first factor's value.
Here's an example:
4 5 8 13 20 29 40 53...
We can regard the first pair of integers, 4 and 5, as the factors "4 x 5." To find their product, count "0, 1, 2, 3, 4" starting with the 4. You land on 20.
Try again with the second pair, "5 x 8." Count from 0 to 5 starting with the 5, and you land on 40.
This works with every pair of integers on every offset curve on the spiral.
As a consequence of this, every integer on a product curve is intersected by a second curve that has the factors explicitly present on it. The intersection of the two curves is like a visual statement of the multiplication: the product lies on one curve and the factors lie on the other.
The product in this example is 24. The blue product curve is S–1, indicating a difference of 2, so the factors are 4 x 6. Those factors appear in order on the red factor curve. Counting up from 4 on the red curve (starting with zero), we arrive on the count of 4 at 24 on the blue curve. It all works out neatly.
But that's not all. There is a relationship between the adjacent integers on one curve and the adjacent integers on the other. On one curve those numbers are 15 and 35; on the other, 16 and 34. The difference between corresponding numbers is one.
The intersection of the two curves seems to contain a great deal of information in geometric form about the factorization.
Nice collection of myths about Arianrhod, Lady of the White Book, Keeper of the Spiral Castle:
Arianrhod appears in the Mabinogion, known originally as the White Book of the Rhydderch, as the sister of Gwydion ap Math ap Mathonwy. (Matthews: 25) They are both the children of Don (ibid.) The story says that when Math lost his footholder, it was Gwydion who elected to have his sister taker her place. (ibid.) The footholder of Math was supposed to be chaste, and so Arianrhod was put to the test. (ibid.) The test consisted of her having to step over the wand of Math. When she did so, she prematurely gave birth to twin sons: Lleu and Dylan. (ibid.)
After Arianrhod gave birth to the twins, Dylan and Lleu, Dylan is said to have been drowned. In some local stories, it is either Math or Arianrhod who throws Dylan into the sea. (Coulter and Turner: 68) In the Mabinogion, it states that, "He took on its nature, and swam as well as the best fish that was therein." Moreover, for that reason he was called Dylan, meaning "Son of the Wave." (Spence: 27) It was said that beneath him no wave ever broke. Another legend states that Dylan was killed from a spear by his uncle Gobhan, and that the waves of the surrounding lands wept for him. (ibid.) The sound of the sea running up the Conway River is still called, "Dylan's Death Groan." (ibid.) This story appears to be very reminiscent of the slaying of the Irish goddess Brigit's son, Ruadan, who was killed by a single blow from his uncle Gobhan's spear. Near Glynllifon, a place that has local folklore concerning Arianrhod, there is a place refereed to by the men who love there as Pwynt Maen Dylan. (Rhys: 210)
Medrawd can be connected to the Irish Ruadan or the earlier Welsh Dylan. According to Rolleston in his book, "Celtic Myths and Legends," he specifically shows Medrawt as being the equivalent of Dylan and Later becoming Sir Mordred of Arthurian legend. (Rolleston: 352) In the Annales Cambriae, Medrawd is said to have died at Camlon in 539 AD.
Caer Arianrhod is the name given to the Corona Borealis and to the sea-laved castle of the same name which is said to be located near the pre-historic mound of Dinas Dinlle near the Menai Straits. (Rhys: 645)
The reef or submerged rock, off the west coast of Arvon is often referred to as Caer Arianrhod. (Spencer: 27) John Rhys suggests that Arianrhod may have been a water faerie who lived in a water-girt castle. (ibid.) It is suggested by Lhuyd that this submerged rock may have been referred to as Caer Arianrhod since the period of the Mabinogion itself. (ibid.) Modernly, Caer Arianrhod is referred to as Caranthreg. (Rhys: 207)
Arianrhod means "silver circle" and may be a reference to the flow of the sea around the submerged rock. (Spence: 27) The Welsh Triads refer to Arianrhod by saying, "Round her flows the River Efnys." (ibid.)
In Llangefni, near Anglesey, there are stories about women who came from Tregar Anthreg to Caer Loda to fetch food or water, and looking back, they saw a town which had been flooded by the sea and that the walls could still be seen at low water. (Rhys: 207) One of the women was called Gwennan, and she was buried at Bed Gwennan. (ibid.) Tregar Anthreg can easily be witnessed from Dinas Dinlle as a rock in the water that is located not too far from the shore. (ibid.)
Another variation of the same story relates how three sisters by the names of Gwen, Elan, and Maelan came from Tregan Anhreg to gather provisions and while they were gone their own city was consumed by the sea. (Rhys: 208) Gwen ran to Bed Gwennan. Maelan fled to Rhos Maelan, otherwise known as Maelon's Moor. Elan went to Tydyn Elan which is also called Elan's Holding. All of these locations are names of places in the nearby area of Anglesey. (ibid.) In the area of Glynllifon, the same story is told, but the title of bi don is attached to the end of each of the three sister's names. (ibid.) Bi don appears to be a late period method of saying "child of Donn." (Rhys: 210)
Within the area surrounding Glynllifon, the place identified as Caer Arianrhod was though to be a place of wickedness. For this reason, only Arianrhod's sisters were permitted to escape. (Rhys: 209) Arianrhod was believed to have drowned, but there was no reason for her death ever supplied in the local stories.(ibid.)
Lhuyd suggests that Arianrhod became the Arthuran Argentem or Queen of Avalon, who is referred to elsewhere as Morgan le Fee' and described by Layamon, and early English poet. (Spence: 27)
In Brittany, the term morgan is used to refer to a mermaid. (Spence: 28) Welsh morgans are lake faeries who have a love for deep water. (Tongue: 26) In some parts of Wales, the morgan is said to kidnap children. (Spence: 28)
Within Welsh culture, the significance and view of lakes and the sea are the same, and so the morgan is a water faerie who can easily be found in either location. (Tongue: 28) Morgan, or Morgen, and the older form of the word Morien, means "sea-born" or "offspring of the sea." (Rhys: 373) It is directly related to the Irish Muirgen which was an epithet for a lake lady in Ireland by the name of Liban. (ibid.) The story of Liban relates how she was neglectful in the covering of a well and consequently the waters rushed forth and became Lough Neagh. (ibid.) Liban did not die, but rather made her home beneath the lake until she changed into a salmon. (ibid.) Morgan is also the name of the lady of the lake who cares for Arthur at her home in Avalon. She is also the half-sister of Arthur. (Rhys: 374)
I found a copy of "Donald in Mathmagic Land" on YouTube...
Posted by Jon Rubin at June 25, 2006 07:42 PM
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