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	Ulam spiral
		You can also see the position (x, y) in the spiral and the number n of any point of the graph by moving the cursor to that point.
		Change the starting number in the center of the spiral by typing a new number (up to 14 digits) in the right input box and press the return key.
		For instance, when we select the center of the spiral equal to 41 by entering this number in the right input box, we will find a diagonal with a lot of primes in the direction NW-SE.
	www.alpertron.com.ar /ULAM.HTM (452 words)

	Ulam spiral
		The Ulam spiral, or prime spiral (in other languages also called Ulam cloth) is a simple method of graphing the prime numbers that reveals a pattern which has never been fully explained.
		It was discovered by the mathematician Stanislaw Marcin Ulam in 1963, while doodling on scratch paper at a scientific meeting.
		This was so significant, that the Ulam spiral appeared on the cover of Scientific American in March 1964.
	www.ebroadcast.com.au /lookup/encyclopedia/ul/Ulam_spiral.html (261 words)

	Ulam spiral
		During a boring meeting one day in 1963, Ulam drew a square, marked the number 1 at the center, and then wrote the increasing whole numbers as spiral that wound its way out to the edge of the paper.
		In Ulam's words the arrangement of primes "appears to exhibit a strongly nonrandom appearance." Ulam rushed home and expanded the spiral to cover a much larger portion of the number sequence.
		The Ulam Spiral should perhaps be known as the "Clarke Spiral" in view of the fact that Arthur C. Clarke described the phenomenon in his novel The City and the Stars (1956, Ch.
	www.daviddarling.info /encyclopedia/U/Ulam_spiral.html (413 words)

	Ulam (Site not responding. Last check: )
		Ulam's Spiral, (sometimes known as "Ulam's Cloth"), is a graphical representation of the natural positive integers which exhibits some remarkable patterns amongst the prime numbers.
		Ulam's Spiral comprises a square pattern where the length of each side increases by 2, (from an initial length of 3), with each rotation of the spiral.
		The spiral is traditionally drawn anti-clockwise with number 1 in the centre and number 2 appearing at the 3 O'Clock position.
	www.apophenia.freeuk.com /ulam.htm (380 words)

	Reference.com/Encyclopedia/Ulam spiral
		It was discovered by the mathematician Stanisław Ulam in 1963, while doodling on scratch paper at a scientific meeting.
		Ulam, bored that day, wrote down a regular grid of numbers, starting with 1 at the center, and spiraling out:
		In the Ulam spiral adjacent diagonals are alternatively odd and even numbers, in both ways.
	www.reference.com /browse/wiki/Ulam_spiral (426 words)

	Math Trek: Prime Spirals, Science News Online, May 4, 2002 (Site not responding. Last check: )
		One particularly intriguing approach is to arrange the integers in a square spiral, starting with 1 at the center of a grid of squares, then mark the squares containing primes.
		Checking all such numbers less than 10 million, Ulam and his coworkers found the proportion of primes to be 0.475.
		Illustrations of prime spirals and other representations of the distribution of prime numbers can be found at http://www.cs.unh.edu/~charpov/Programming/PostScript-primes/, http://platon.lacitec.on.ca/~dmorin/applet/village/, and http://yoyo.cc.monash.edu.au/~bunyip/primes/.
	www.sciencenews.org /20020504/mathtrek.asp (1065 words)

	Inquiry Example #1 (Site not responding. Last check: )
		Ulam's Spiral was discovered by mathematician Stanislav Ulam during a boring lecture.
		Ulam began by drawing a grid of integers spiralling out from 1 in the middle and shading in all the primes:
		Mathematicians have still not been able to construct a "prime-generating function"; maybe an investigation of Ulam's spiral could produce such a function.
	teachers.crescentschool.org /weissworld/m3/spiral/spiral.html (529 words)

	AS/SA No 14, Article 7: Peter Marteinson, "Observations on the Regularity of Prime Distribution"
		Conversely, it can be demonstrated that each of the three even series on the spiral can be generated by some combination of two primes, either both in the five o'clock series, both in the seven o'clock series, or one in each, without exception, using simple modular arithemtic.
		Finally, it would seem clear from figure 4 that Ulam's spiral displays a strikingly "non-random" appearance only insofar as it is essentially a geometric distortion of the highly regular modulus-6 spiral I have drawn.
		But to me the most interesting aspect of this way of looking at primes and their distribution is that it provides a simple and elegant way to grasp why certain numbers lack factors and a thus clear understanding of the nature of primes.
	www.chass.utoronto.ca /french/as-sa/ASSA-14/article7en.html (1687 words)

	Prime Number Spiral 6.44 - Free Download. Software for exploring the Prime Number Spiral (a.k.a. the Ulam Spiral).
		the Ulam Spiral) is constructed as follows: Consider a rectangular grid.
		We start with the central point and arrange the positive integers in a spiral fashion (anticlockwise).
		There is a tendency for the prime numbers to form diagonal lines, however far out into the spiral one goes.
	www.gold-software.com /download4993.html (155 words)

	Ulam's Spirals : Towards an Explanation
		It is good to observe that over a good range of k values say from 1 to 999 the mean value of all multiplication factors approaches 2 (which represents the multiplication factor for all the odd numbers together that contribute to build all those diagonals.
		To further illustrate the "predictive" Ulam's diagonal prime density computation, table5 has been developped : it shows for selected values of k the observed/real number of prime terms among the 100 first terms, i.e.
		The "predicted" number appearing besides the "observed" number is obtained by summing the "predictive prime density" applying to each term, as shown in the above example.
	www.geocities.com /dhvanderstraten/ulamtxt.html (1505 words)

	Ulam's Rose (prime number spiral)
		Stanislaw Ulam was attending some boring meeting, and to divert himself somewhat he began to scribble on a piece of paper.
		Ulam ran home and expanded the spiral to cover a much larger portion of the number sequence.
		The Ulam Rose of 1 => 262,144 used here is an embellishment of an image originally created by Jean-François Colonna ©1996, CNET and the École Polytechnique, Paris France.
	www.abarim-publications.com /artctulam.html (731 words)

	The mysterious Ulam spiral phenomenon (Site not responding. Last check: )
		This was accidentally discovered by nuclear physicist Stanislaw Ulam while he was passing the time during a boring lecture.
		Prime Number Spiral: P. Meyer's software for exploring the Ulam spiral phenomenon in the distibution of primes.
		S.M. Ellerstein, "The Pronic Renaissance: The Ulam Square Spiral (Modified)" Journal of Recreational Mathematics 29 (3), 1998, p.
	secamlocal.ex.ac.uk /people/staff/mrwatkin/zeta/ulam.htm (339 words)

	NumberSpiral.com -- Links
		Towards the bottom of the home page, the author rolls his graph into a three-dimensional coil, stands it on end, and glues it to the number spiral, thereby extending the number spiral into a third dimension.
		Ulam's Spirals: Towards an Explanation by Didier van der Straten.
		Among the topics covered is the Ulam Spiral.
	www.numberspiral.com /p/links.html (279 words)

	Ulam Spiral
		The Ulam spiral, named for Stanislaw Ulam, seems to show a large number of continuous diagonal 'strings' for prime numbers (2 divisors).
		) an interesting spiral appears when the start number is a square and all other plotted points along the Ulam spiral are squares.
		I believe that this method of coiling whole numbers (The Ulam spiral) allows for a myriad of interesting patterns and has no significance to Prime numbers, it just happens.
	home.netcom.com /~abraxas2/Ulam.htm (448 words)

	NumberSpiral.com -- Primes
		As shown in Figure 1, the densest concentrations of primes seem to occur mainly in curves whose offsets are prime.
		Each diagonal of the Ulam spiral corresponds to a particular curve on the number spiral.
		However, the Ulam spiral makes two diagonals out of each curve by allowing both ends of a diagonal to grow in opposite directions and placing alternate members of the sequence at either end.
	www.numberspiral.com /p/p004.html (308 words)

	The mysterious Ulam spiral phenomenon
		This was accidentally discovered by nuclear physicist Stanislaw Ulam while he was passing the time during a boring lecture.
		Prime Number Spiral: P. Meyer's software for exploring the Ulam spiral phenomenon in the distibution of primes.
		S.M. Ellerstein, "The Pronic Renaissance: The Ulam Square Spiral (Modified)" Journal of Recreational Mathematics 29 (3), 1998, p.
	www.secamlocal.ex.ac.uk /people/staff/mrwatkin/zeta/ulam.htm (339 words)

	Voronoi Diagram of the Ulam Prime Spiral
		In 1963, the mathematician Stanislaw Ulam was sitting, bored, in a lecture.
		I told the applet to draw the Voronoi Diagram of the Ulam Prime Spiral, and obtained the picture below.
		Its (x,y) coordinates in the Ulam Prime Spiral.
	www.dr-mikes-maths.com /psv/index.html (812 words)

	Prime numbers in PostScript
		In that picture, called a Ulam's spiral, natural numbers are placed along a squared spiral and a dot is printed for each prime number, like this (for a mysterious reason, I decided to start my spiral with 1 instead of 0; please don't tell Dijkstra):
		The nice thing is, when you consider a large enough number of primes, you begin to see some patterns and especially alignments appear.
		You can edit the source to change the resolution (in dpi) and to choose to have the lines of the spiral and the numbers appear or not.
	www.cs.unh.edu /~charpov/Programming/PostScript-primes (376 words)

	Ulam Rose? Get A Grip!
		First of all, let's wonder what exactly the Ulam Spiral is. Well, to start with, the Ulam Spiral is not a spiral but isolated segments of the number sequence that get larger 8 numbers per segment:
		Imagine the Amazing Abarim Publications Pillar to be rolled up to a tube so that the number sequence becomes a spiral swirling down that tube.
		You will find that all iso-factors (lines that connect all numbers divisible by some other number) are spirals too, and swirl along the same tube.
	www.abarim-publications.com /ulamgohome.html (816 words)

	Ulam Spiral Programs - Prime Number Spiral, Animated Halloween Wallpaper, MITCalc - Torsion Springs, ...
		The Prime Number Spiral a k a the Ulam Spiral is formed by marking the prime...
		Geometric and strength designs of spiral cylindrical torsion springs loaded w...
		The spiral of terror is unleashing skulls on your desktop.
	www.softchecker.com /files/ulam-spiral.html (516 words)

	Ulam spiral downloads - MITCalc - Springs - 15 types, MITCalc - Torsion Springs, Rhumb Line
		Keywords: conical springs, spring, spiral springs, cylindrical spring
		The calculation is intended for the purposes of geometric and strength designs of spiral cylindrical torsion springs made of wires and rods in circular sections, loaded with a static or cyclic loading.
		The object is to out-navigate the opposition and score the most points to rule the waves.
	www.vicman.net /lib/ulam-spiral.htm (708 words)

	The Ulam Prime Spiral
		Ulam's Spiral provides us with striking visual evidence that there is some order in the distribution of prime numbers.
		In the two first images at top, the prime numbers are plotted with a white pixel.
		Here the first 27 Mersenne primes are plotted by the spiral method...
	yoyo.cc.monash.edu.au /~bunyip/primes/primeSpiral.htm (151 words)

	Ulam Spiral: Prime Number Spiral, 3D SurfTris, stairs48.exe, ...
		Ulam Spiral: Prime Number Spiral, 3D SurfTris, stairs48.exe,...
		the Ulam Spiral) is formed by marking the prime numbers in a spiral arrangement of the natural numbers.
		With IconsManager you can arrange the icons on the desktop at the left, right, top, bottom, left and right, top and bottom, in a spiral etc. IconsManager will help you to group the icons in the categories and place them on the separate virtual desktops.
	www.fileedge.com /all/ulam-spiral (430 words)

	Ulam Spiral Software: Prime Number Spiral, JavaScript Spiral PopMenu, Fear Spiral Animated Screensaver, MITCalc - ...
		The Prime Number Spiral a k a the Ulam Spiral is formed by marking the prime numbers in a spiral arrangement of the natura...
		Creates professional menus with spiral effects for web pages With this easy to use web menu development tool you can creat...
		Geometric and strength designs of spiral cylindrical torsion springs loaded with static or cyclic loading Application supp...
	www.windfile.com /find/ulam-spiral (364 words)

	Prime Number Spiral
		An interesting prime number spiral was discovered in 1963 by Stanislaw M. Ulam, and is now called "the Ulam spiral".
		We start with the central point and arrange the positive integers in a spiral fashion (anticlockwise) as at below left.
		Languages: Prime Number Spiral is bilingual: English and German.
	www.hermetic.ch /pns/pns.htm (372 words)

	Best free ulam spiral downloads. Software for exploring the Prime Number Spiral (a.k.a. the Ulam Spiral) Tetris clone ...
		the Ulam Spiral) Tetris clone with playing field spread upon 5 different 3D surfaces.
		Ulam spiral software: For exploring the Prime Number Spiral, 48-piece jigsaw puzzle of some spiral stairs, Tetris clone with 3D playing field and more.
		the Ulam Spiral) is formed by marking the prime numbers in a spiral arrangement of the natural numbers.
	www.freedownloadmanager.org /downloads/ulam_spiral_software (95 words)

	[No title] (Site not responding. Last check: )
		Some clips from a wonderfully-done exploration of this curious mathematical construct (also known as the Ulam Spiral).
		To date, it is not fully understood exactly why more primes tend to fall along certain "curves" in the spiral than others, but the elusive quest for a simple prime-generating formula keeps the deceptively-simple spiral intriguing to number theorists.
		btw, did you know that "the spiral" had always been an universal symbol of "generative force", a kind of "life source" and thus "mother nature", as well as the "universal order", according to Marija Gimbutas' research on the iconography of Early Neolitihic settlements of Europe?
	www.clipmarks.com /clipmark/3070A6D4-B7A4-4EC1-A1A6-0DC57973763F (178 words)

	Mathematica - simplifying ulam spiral code
		Ulam's spiral appeared in the second issue of The Mathematica Journal,
		Draw the spiral and display those with prime index as dots:
		the idea is that the length of the limbs of the spiral follow the
	www.codecomments.com /archive382-2005-4-471100.html (2346 words)

	ulam prime spiral - data visualization & visual design - information aesthetics
		ulam prime spiral - data visualization & visual design - information aesthetics
		a simple line & dot plot, invented by Polish-American mathematician Stanislaw Ulam, in which the positive integers are arranged in a spiral with the prime numbers highlighted in some way along the spiral.
		It doesn't seem to be a special thi...
	infosthetics.com /archives/2006/09/ulam_prime_spiral.html (222 words)

	NumberSpiral.com - Home
		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:
		The following number spiral shows all the primes that occur within the first 46,656 non-negative integers.
	www.numberspiral.com (301 words)

	Opt/imization and Vis/ualization in Mathematics - Ulam Prime Spiral
		Opt/imization and Vis/ualization in Mathematics - Ulam Prime Spiral
		An image representation of the Ulam Prime Spiral can be generated in three lines of Matlab code, where
		More information on Ulam's spiral and prime numbers can be found here:
	optvis.com /content/view/1/40 (334 words)

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