1. Introduction
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:
If we continue winding for a while and zoom out a bit, the result looks like this:
If we zoom out even further and remove everything except the dots that indicate the locations of integers, we get the next illustration. It shows 2026 dots:
Let's try making the primes darker than the non-primes:
The primes seem to cluster along certain curves. Let's zoom out even further to get a better look. The following number spiral shows all the primes that occur within the first 46,656 non-negative integers. (For clarity, non-primes have been left out.)
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.
On the next few pages of this website, we'll investigate these patterns and try to make sense out of them.
Details
Each non-negative real number n has polar coordinates
where theta is the angle measured in rotations, not radians or degrees. One rotation equals 360 degrees.
Unless otherwise noted, all angles on this website are measured in rotations. The reason for this will become clear as we go on.
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:
If we continue winding for a while and zoom out a bit, the result looks like this:
If we zoom out even further and remove everything except the dots that indicate the locations of integers, we get the next illustration. It shows 2026 dots:
Let's try making the primes darker than the non-primes:
The primes seem to cluster along certain curves. Let's zoom out even further to get a better look. The following number spiral shows all the primes that occur within the first 46,656 non-negative integers. (For clarity, non-primes have been left out.)
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.
On the next few pages of this website, we'll investigate these patterns and try to make sense out of them.
Details
Each non-negative real number n has polar coordinates
where theta is the angle measured in rotations, not radians or degrees. One rotation equals 360 degrees.
Unless otherwise noted, all angles on this website are measured in rotations. The reason for this will become clear as we go on.
1 comment:
Many 카지노사이트.online patrons do not realize that they can get kicked out for making an attempt to capitalize on another person's winnings—even if that person has deserted their steadiness on a slot machine. Michael Page, who works at San Manuel Indian Bingo & Casino, mentioned in a special Quora thread that that is generally referred to as "ticket browsing." Buffalo™ Gold offers gamers an enhanced expertise to their favorite Buffalo™ recreation. Choose among 24 exciting table games, with friendly sellers and an atmosphere of unparalleled consolation. Try your hand at such favorites as Blackjack or Roulette, or take a shot at Mini-Bacc, Craps, Cajun Stud, Head’s Up Hold’em, Flop Poker and High Card Flush. If you’re in the temper for exciting, high-stakes action, our dedicated excessive restrict table games are is the place.
Post a Comment