From: 3blue1brown
The patterns observed when plotting prime numbers in polar coordinates reveal insights into rational approximations for pi [00:00:15]. This visualization method, where points are labeled with a distance from the origin (radius r) and an angle (theta) measured in radians, provides a unique way to understand these mathematical relationships [00:00:25]. In radians, an angle of pi (π) represents halfway around a circle, and 2 pi (2π) represents a full circle [00:00:44]. Adding 2π to the angle coordinate does not change the point’s location [00:00:51].
Visualizing Approximations
When plotting points where both the radius and angle are a given whole number (e.g., (1,1), (2,2), (3,3)), these points spiral outwards, forming an Archimedean spiral [02:01:00]. Each step forward increases the angle by one radian and the distance by one unit [01:56:00]. The emergence of distinct spiral arms at different scales is directly tied to how closely certain integers approximate a multiple of 2π.
The 6-Radian Approximation
At a smaller scale, six distinct spiral arms can be observed in the plot of all whole numbers [04:36:00]. This occurs because six radians is slightly less than a full turn (2π radians) [05:07:00]. Therefore, every sixth point in the sequence returns to approximately the same angular position, creating the illusion of a continuous curving line [05:24:00]. Each of these spiral arms corresponds to a “residue class mod 6” [06:52:00].
The 22/7 Approximation for Pi
Zooming out further, a different pattern emerges with 44 distinct spirals [03:54:00]. This phenomenon is due to the fact that taking 44 steps, turning 44 radians, is very close to a whole number of turns [07:17:00]. Specifically, 44 radians equates to 44 divided by 2π rotations, which is just over 7 full turns (approximately 7.0069 rotations) [07:26:00].
This implies that 44/7 is a close approximation for 2π, which is more commonly recognized as the famous 7 approximation for pi [07:37:00]. Because 44 is very close to 7 * 2π, when counting up by multiples of 44, each point has almost the same angle as the last, resulting in a very gentle spiral that appears as a distinct arm [07:57:00]. Each of these 44 spiral arms represents a residue class mod 44 [08:22:00].
The 355/113 Approximation for Pi
At an even larger scale, the spirals give way to outward-pointing rays [02:40:00]. This is explained by an even more precise rational approximation for pi: 355/113 [00:11:38]. This approximation is unusually good [00:11:53].
The visual effect comes from the fact that 710 radians (which is 2 * 355 radians) is extremely close to a whole number of full turns [00:11:06]. Performing the calculation, 710 radians divided by 2π rotations equals approximately 113.000095 rotations [00:11:19]. This means that when moving forward by steps of 710, the angle of each new point is almost exactly the same as the last, just microscopically bigger [00:12:07]. Consequently, these sequences appear as nearly straight lines, which are actually very gentle spirals [00:12:15]. The fact that it takes so long for the spiral to become prominent is a testament to how good this approximation for 2π is [00:12:48].
Unusually Good Approximations
These specific rational approximations for pi (22/7 and 355/113) are considered “unusually good” because they are significantly more accurate than one might expect for other famous irrational numbers like phi or the square root of 2 [00:11:50]. Understanding the origin of such approximations and what makes them unusually good often involves studying concepts like continued fractions [00:11:50].
Relation to Prime Numbers
While the patterns of spirals and rays are primarily explained by these rational approximations for pi, the initial visualization specifically involved prime numbers [00:01:04]. The absence of certain spiral arms or rays when only primes are plotted is due to divisibility rules [00:08:33]. For example, primes cannot be multiples of 44 (except for 2 and 11), nor can they fall into residue classes that share common factors with 44 (e.g., even numbers or multiples of 11) [00:08:33]. This filtering effect of prime numbers reveals the underlying structure created by the approximations [00:09:01].
Euler’s totient function (phi, φ) quantifies the number of integers coprime to a given number ‘n’ (i.e., not sharing any prime factors with n) [09:56:00]. For example, φ(44) = 20, which corresponds to the 20 visible spirals when filtering for primes using the 44-radian approximation [09:43:00]. Similarly, for the 710-radian approximation, the number of visible rays (280) is equal to φ(710) [00:14:04]. The distribution of primes within these remaining residue classes is described by Dirichlet’s theorem on arithmetic progressions [00:17:27].