Researchers thought this was a bug (Borwein integrals)
TLDRThe video script explores the intriguing pattern of computations involving the sinc function, which equals pi, and its surprising stability before a subtle deviation. It introduces the concept of moving averages and relates them to integrals through Fourier transforms and the convolution theorem. The explanation reveals how complex mathematical problems can be simplified by shifting perspectives, hinting at the power of these tools in both understanding and solving intricate mathematical phenomena.
Takeaways
- π The universe sometimes presents patterns that seem random but are actually predictable, as illustrated by a sequence of computations that all equal pi.
- π The script introduces the sinc function (sine of x divided by x), which is significant in math and engineering, and has a unique property when integrated from negative infinity to infinity, equaling pi.
- π€ The mystery deepens as modifications to the sinc function, such as stretching and multiplying, seem to not affect the integral's value of pi, until a certain point.
- π The value of pi is stable throughout the sequence until it is barely less than pi at a specific point, which is not due to numerical error but a real phenomenon.
- π¨βπ¦ The pattern was discovered by Jonathan and David Borwein, who noted the surprising stability and the eventual tiny deviation from pi in their research.
- π The script introduces a seemingly unrelated sequence of functions that mimic the stable pattern observed in the sinc function sequence, using moving averages.
- π The moving average process is analogous to the integrals of the sinc function, showing a stable value until a certain iteration where the plateau becomes thinner than the window width.
- π Fourier transforms and convolutions are hinted as the key to understanding the connection between the sinc function and the moving average sequence.
- π§ The convolution theorem, in particular, is teased as a powerful tool that not only explains the observed patterns but also has practical applications like fast computation of large numbers' products.
- π The video script serves as a teaser for a deeper dive into the convolution theorem and its implications, promising a better understanding of these mathematical tools.
Q & A
What is the main topic of the video script?
-The main topic of the video script is the exploration of a mathematical phenomenon related to the sinc function and its integral, which evaluates to pi, and the subsequent investigation into the pattern and behavior of a sequence of computations involving the sinc function.
What does the sinc function represent?
-The sinc function represents a normal oscillating sine curve that is squished down as it gets farther away from zero by multiplying it by 1 over x. It is commonly used in math and engineering and is often redefined at 0 to equal 1 for continuity.
Why is the integral of the sinc function from negative infinity to infinity exactly pi?
-The integral of the sinc function from negative infinity to infinity evaluates to pi because it represents the area between the curve and the x-axis, taking into account both positive and negative parts of the graph. This is a well-known result in mathematics, although the explanation for why it occurs is not immediately clear using standard calculus tools.
What happens when the sinc function is modified and multiplied by increasingly larger odd numbers?
-When the sinc function is modified by stretching it horizontally by increasingly larger odd numbers and then multiplied, the resulting wave becomes more complex with its mass concentrated towards the middle. Despite these modifications, the integral of these modified functions continues to evaluate to pi, which is a mystery that the video aims to explain.
What is the significance of the value 15 in the sequence of computations?
-The value 15 is significant because it is the point at which the sequence of computations stops evaluating to pi and instead gives a value that is just barely less than pi. This occurs because the sum of the reciprocals of the odd numbers used in the sequence becomes larger than 1 at this point, causing the plateau in the corresponding moving average function to disappear entirely.
How does the video script relate the sequence of computations to a seemingly unrelated phenomenon?
-The video script relates the sequence of computations to a seemingly unrelated phenomenon by introducing a sequence of functions that mimic moving averages. The values of these functions at the input 0 initially remain stable (equal to 1) until the width of the plateau becomes thinner than the window used for the moving average, at which point the value falls just short of 1. This phenomenon is analogous to the integrals in the sinc function sequence, where the value remains stable (equal to pi) until it breaks down at a certain point.
What is the role of Fourier transforms in explaining the mathematical phenomenon discussed in the script?
-Fourier transforms play a crucial role in explaining the mathematical phenomenon by providing a new perspective on the sinc function. The Fourier transform of the sinc function is related to the rect function, which is a top hat function. This transformation allows for the computation of the integral of the sinc function to be rephrased as evaluating the rect function at zero, which simplifies the problem significantly.
What is the convolution theorem mentioned in the script, and how does it relate to the problem at hand?
-The convolution theorem is a mathematical principle that states that the Fourier transform of the product of two functions is equal to the convolution of their Fourier transforms. In the context of the script, this theorem is used to explain why multiplying sinc functions corresponds to a series of moving averages, which in turn helps understand the stable value observed before the breakdown in the sequence of computations.
How does the script use the concept of moving averages to explain the sinc function sequence?
-The script introduces a sequence of functions that act as moving averages of the previous function. Each function in the sequence is defined by taking the average value of the previous function within a sliding window of increasing width. This process creates a plateau that shrinks with each iteration. The value of these functions at the input 0 remains stable until the plateau becomes thinner than the window, at which point the value falls just short of 1. This is analogous to the stable value in the sinc function sequence before it breaks down.
What is the significance of adding the 2 cosine of x term in the integral?
-Adding the 2 cosine of x term in the integral extends the pattern of the stable value before it breaks down. This is because the resulting function has a longer plateau, which takes more iterations of the moving average process to shrink completely, corresponding to the number 113 in the sequence.
How does the video script conclude in terms of the power of Fourier transforms?
-The video script concludes by highlighting the power of Fourier transforms as a systematic way to shift perspective and make difficult problems appear easier. It suggests that understanding these powerful tools, such as the convolution theorem, can provide valuable insights and even lead to efficient algorithms for complex computations, such as quickly computing the product of two large numbers.
Outlines
π The Mysterious Pattern of Pi in Sine Computations
This paragraph introduces a sequence of computations that all unexpectedly equal pi, despite appearing to follow a random pattern. The focus is on the sinc function, which is the sine of x divided by x, and its integral from negative to positive infinity, which equals pi. The speaker aims to explain this phenomenon and its relation to a seemingly unrelated pattern observed in a different mathematical context.
π Moving Averages and the Function Sequence
The second paragraph delves into a sequence of functions starting with a simple step function called f1(x), which is then averaged with a sliding window to create subsequent functions. The speaker highlights how the plateau of each function in the sequence shrinks with each iteration and how this relates to the stability and eventual break from the value of 1. The analogy is drawn to the integrals discussed earlier, where a stable value of pi is maintained until a certain point is reached.
π’ Breaking Down the Sequence at 113
In this paragraph, the speaker explains that the sequence of computations and the corresponding moving averages both break down at the number 113, which is not a coincidence. The connection between these two seemingly different sequences is explored, hinting at the role of Fourier transforms and convolutions in understanding this phenomenon. The paragraph sets the stage for a deeper exploration of these concepts in future discussions.
π Fourier Transforms and Convolution Theorem
The final paragraph introduces Fourier transforms and the convolution theorem as the keys to understanding the mysterious patterns observed earlier. The sinc function and the rect function are related through a Fourier transform, and the integral of the sinc function is equivalent to evaluating the rect function at zero. The convolution theorem is mentioned as the link between the product of functions and their Fourier transforms, suggesting a new way to view the multiplication of sinc functions as a series of moving averages.
Mindmap
Keywords
π‘sinc function
π‘integral
π‘pi (Ο)
π‘Fourier transform
π‘convolution theorem
π‘moving average
π‘sequence of computations
π‘rect function
π‘numerical error
π‘floating-point arithmetic
π‘Borwein's paper
Highlights
The universe sometimes appears to follow a random yet predictable pattern, as seen in a sequence of computations that all equal pi.
The function sine of x divided by x, also known as sinc, is commonly used in math and engineering and has a unique property where its integral from negative infinity to infinity equals pi.
The sinc function's behavior near x equals 0 is such that it approaches 1, and by redefining it at 0 to equal 1, we get a continuous curve.
A sequence of computations involving the sinc function and stretching it by odd numbers results in integrals that equal pi until a certain point, then deviate by an incredibly small amount.
The paper by Jonathan and David Borwein discusses this phenomenon, which was initially thought to be a bug due to numerical errors but is actually a real occurrence.
Adding a factor like 2 cosine of x to the integrals extends the sequence of pi values until a break at the number 113, demonstrating a surprising stability.
The phenomenon of the sinc function and its integral can be analogized to a sequence of functions defined by moving averages, where a stable value is maintained until a critical point is reached.
The moving average process involves a function that equals 1 between -1 and 1 and 0 elsewhere, with subsequent functions being averages of the previous function over sliding windows.
The length of the plateau in each function of the moving average sequence decreases with each iteration, eventually leading to a break at the width of 1/15.
The moving average sequence demonstrates a stable pattern that breaks down when the sum of the reciprocals of odd fractions exceeds 1.
The analogy between the moving average sequence and the sinc function integrals is not only qualitatively similar but quantitatively the same.
The Fourier transform is introduced as a tool to relate the sinc function with the rect function, providing a new perspective on the problem.
The convolution theorem, which relates the Fourier transform of a product of functions to the convolution of their individual transforms, is key to understanding the behavior of the sinc function sequence.
The convolution operation for the rect function resembles a moving average, which explains the stable value before the breakdown in the sequence.
The convolution theorem has practical applications, such as an algorithm for quickly computing the product of two large numbers.
Fourier transforms provide a systematic way to shift perspective, making difficult problems appear easier to solve.
Transcripts
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