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Euler spiral - Wikipedia, the free encyclopedia

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Euler spiral - Wikipedia, the free encyclopedia

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Euler spirals have applications to diffraction computations. They are also widely used as transition curves in railroad engineering/ highway engineering for connecting and transiting the geometry between a tangent and a circular curve. A similar application is also found in photonic integrated circuits.

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Fermat's spiral - Wikipedia, the free encyclopedia

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Fermat's spiral - Wikipedia, the free encyclopedia

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In disc phyllotaxis, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially.

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Collatz Graph: All Numbers Lead to One - Jason Davies

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Collatz Graph: All Numbers Lead to One - Jason Davies

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Here is a graph showing the orbits of all numbers under the Collatz map with an orbit length of 19 or less, excluding the 1-2-4 loop.

Background

The Collatz conjecture is as follows.

The conjecture is that you will always reach 1, no matter what number you start with. At this point, of course, you end up in an endless loop going from 1 to 4, to 2 and back to 1.

Reverse Collatz

Alternatively, we can formulate the conjecture such that 1 leads to all natural numbers, using an inverse relation (see the link for full details).

If we exclude the 1-2-4 loop, the inverse relation should result in a tree, if the conjecture is true.

Credits

Collatz graph generation based on Python code by @TerrorBite. Radial node-link tree layout based on an example in Mike Bostock's amazing D3 library.

Further Reading

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Here is a graph showing the orbits of all numbers under the Collatz map with an orbit length of 19 or less, excluding the 1-2-4 loop. The Collatz conjecture is as follows. The conjecture is that you will always reach 1, no matter what number you start with.

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Mathematicians Discover Prime Conspiracy | Quanta Magazine

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Mathematicians Discover Prime Conspiracy | Quanta Magazine

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Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers - those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.

Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today, Kannan Soundararajan and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits.

This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers. Most mathematicians would have assumed, Granville and Ono agreed, that a prime should have an equal chance of being followed by a prime ending in 1, 3, 7 or 9 (the four possible endings for all prime numbers except 2 and 5).

"I can't believe anyone in the world would have guessed this," Granville said. Even after having seen Lemke Oliver and Soundararajan's analysis of their phenomenon, he said, "it still seems like a strange thing."

Yet the pair's work doesn't upend the notion that primes behave randomly so much as point to how subtle their particular mix of randomness and order is. "Can we redefine what 'random' means in this context so that once again, [this phenomenon] looks like it might be random?" Soundararajan said. "That's what we think we've done."

Prime Preferences

Soundararajan wondered if similarly strange phenomena appear in other contexts. Since he has studied the primes for decades, he turned to them - and found something even stranger than he had bargained for. Looking at prime numbers written in base 3 - in which roughly half the primes end in 1 and half end in 2 - he found that among primes smaller than 1,000, a prime ending in 1 is more than twice as likely to be followed by a prime ending in 2 than by another prime ending in 1. Likewise, a prime ending in 2 prefers to be followed a prime ending in 1.

Soundararajan showed his findings to postdoctoral researcher Lemke Oliver, who was shocked. He immediately wrote a program that searched much farther out along the number line - through the first 400 billion primes. Lemke Oliver again found that primes seem to avoid being followed by another prime with the same final digit. The primes "really hate to repeat themselves," Lemke Oliver said.

Lemke Oliver and Soundararajan's first guess for why this bias occurs was a simple one: Maybe a prime ending in 3, say, is more likely to be followed by a prime ending in 7, 9 or 1 merely because it encounters numbers with those endings before it reaches another number ending in 3. For example, 43 is followed by 47, 49 and 51 before it hits 53, and one of those numbers, 47, is prime.

But the pair of mathematicians soon realized that this potential explanation couldn't account for the magnitude of the biases they found. Nor could it explain why, as the pair found, primes ending in 3 seem to like being followed by primes ending in 9 more than 1 or 7. To explain these and other preferences, Lemke Oliver and Soundararajan had to delve into the deepest model mathematicians have for random behavior in the primes.

Random Primes

Prime numbers, of course, are not really random at all - they are completely determined. Yet in many respects, they seem to behave like a list of random numbers, governed by just one overarching rule: The approximate density of primes near any number is inversely proportional to how many digits the number has.

Despite its predictive power, Cramér's model is a vast oversimplification. For instance, even numbers have as good a chance of being chosen as odd numbers, whereas real primes are never even, apart from the number 2. Over the years, mathematicians have developed refinements of Cramér's model that, for instance, bar even numbers and numbers divisible by 3, 5, and other small primes.

These simple coin-tossing models tend to be very useful rules of thumb about how prime numbers behave. They accurately predict, among other things, that prime numbers shouldn't care what their final digit is - and indeed, primes ending in 1, 3, 7 and 9 occur with roughly equal frequency.

Yet similar logic seems to suggest that primes shouldn't care what digit the prime after them ends in. It was probably mathematicians' overreliance on the simple coin-tossing heuristics that made them miss the biases in consecutive primes for so long, Granville said. "It's easy to take too much for granted - to assume that your first guess is true."

In a similar way, Soundararajan and Lemke Oliver have found that the biases they uncovered in consecutive primes come very close to what the prime k-tuples conjecture predicts. In other words, the most sophisticated conjecture mathematicians have about randomness in primes forces the primes to display strong biases. "I have to rethink how I teach my class in analytic number theory now," Ono said.

At this early stage, mathematicians say, it's hard to know whether these biases are isolated peculiarities, or whether they have deep connections to other mathematical structures in the primes or elsewhere. Ono predicts, however, that mathematicians will immediately start looking for similar biases in related contexts, such as prime polynomials - fundamental objects in number theory that can't be factored into simpler polynomials.

And the finding will make mathematicians look at the primes themselves with fresh eyes, Granville said. "You could wonder, what else have we missed about the primes?"

This article was reprinted on Wired.com.

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Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers - those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.