# The Shape of Pi

## by Jason Padgett

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$10,000

Dimensions

18.500 x 18.000 inches

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Title

The Shape of Pi

Artist

Jason Padgett

Medium

Drawing - Pencil On Paper

Description

This captivating illustration presents Pi, geometrically depicted as 720sin(Pi/720), with a radius of one, as if it were light. Jason meticulously sketched it using pencil and paper, subsequently inverting the colors to enhance visibility from his perspective.

Within this drawing, Jason emphasizes the absence of perfect circles throughout the universe. Circles are inherently composed of triangles, just like all other shapes, and are profoundly intertwined with the number Pi. The closer you cram triangles within a given perimeter, the more the resulting shape approximates a circle, yet never truly attains it. This elucidates why Pi extends to infinity. Pi represents the circumference of a circle (the distance around its perimeter) divided by its diameter (the distance from one side, through the center, to the opposite side). With the addition of more triangles, the circumference incrementally expands (though by diminishing increments), while the diameter remains constant. For instance, a circle comprising 180 right triangles yields a value of 3.141433 when its circumference is divided by its diameter (accurate to three decimal places). Similarly, a circle consisting of 360 right triangles (one right triangle per degree) results in 3.141552 (accurate to four decimals). The present image encompasses 720 right triangles, producing a value of 3.141582 when dividing its circumference by its diameter (accurate to four decimals, nearly reaching five). If we were to draw a circle with 25,000 right triangles (as nature does and even surpasses), the value would be 3.141592645 (accurate to seven digits). Thus, these values precisely describe specific shapes. Specifically, 720sin(Pi/720) represents this image exclusively, denoting a distinct ratio and shape. Consequently, as Pi extends numerically, it characterizes particular shapes that progressively approximate a circle but never achieve it. Therefore, the circumference of a "circle" with 1440 sides is not PiR^2 but rather 1440sin(Pi/1440)R^2. Although the disparity between the two is small, precise measurements become crucial at the quantum level. Padgett's method ensures flawless measurements, devoid of error.

Pi's astonishing quality lies in its all-encompassing nature. It permeates everything in the universe. All shapes, including circles (which, in reality, are not perfect circles but rather composed of triangles), disintegrate into straight lines and triangles. Every curve can be deconstructed into infinitesimal straight lines that subtly change in angle (similar to this drawing, which appears curved but comprises perfectly straight lines). In mathematics, specifically trigonometry, there exists a method to convert measurements from degrees (360 degrees constituting a circle) to measurements using Pi (referred to as Pi radians). To achieve this conversion, one multiplies the degrees by Pi/180 (the fraction representing Pi divided by 180). A straight line always encompasses 180 degrees (two right triangles forming a straight line: 90 + 90 = 180). Likewise, any triangle's three internal angles, when combined, always sum up to 180 degrees (equivalent to a straight line). Since all existing shapes can be deconstructed into straight lines and triangles, let's observe what occurs when degrees are converted into Pi radians. Multiplying 180 by Pi/180 yields 180Pi/180 (or, in words, 180 times Pi divided by 180). The two instances of 180 cancel each other out, as 180 divided by 180 equals one, resulting in Pi! Therefore, every shape in the universe can be broken down into fragments of Pi, quite literally. Nature truly possesses a sense of humor!

This illustration solely employs straight lines. It comprises 720 right triangles, each rotating by 1/2 degree. Although there is no actual curve, the accumulation of triangles along the perimeter creates the illusion of a circle or curve. A mountain serves as an analogy: when viewed up close, it appears non-circular; however, when seen from outer space, the Earth seems perfectly spherical. Consequently, what constitutes a circle is relative. From an earthly perspective, the mountain lacks circularity, whereas from the vantage point of outer space, it resembles a circle (or part thereof). This exemplifies one of the numerous captivating illusions of relativity. For those fond of equations, graphing f(x) = xsin(Pi/x) demonstrates the outcome. As x (representing the number of sides in a circle) approaches infinity, f(x) gradually approaches Pi.

Lastly, the crucial revelation about Pi is that we have been taught it is endless. While this holds true in a pure equation or mathematical context, it is inaccurate when considering a "circle" in reality. This is the truly astounding aspect: Pi physically terminates or, more descriptively, concludes in an observational sense. The smallest measurable unit in the universe is known as the Planck length, which emerges from the limitations of observation. To perceive an object, photons are bounced off it, and by analyzing the resulting geometry, we gain insights into what is being observed. For finer readings or smaller objects, more photons (or greater energy) must be used during observation. However, there exists a physical "limit" to observation. When attempting to observe something smaller than the Planck length, such an object must be bombarded with such immense energy that it collapses into a black hole, absorbing the photons and leaving nothing observable. Although points smaller than the Planck length exist, they elude our observation and therefore lack "relativity." This concept was discovered by Max Planck, who was awarded the Nobel Prize for his findings. The Planck constant is universally accepted and utilized by mathematicians, nuclear physicists, and scientists from various disciplines. Consequently, physically adding an infinite number of triangles within a "circle" is impossible. The addition of triangles can only continue until each microscopic side of the circle measures one Planck length, representing the point where Pi physically (or relativistically) concludes without question. Mathematicians may have overlooked this insight (in my opinion) due to the absence of viewing mathematics from a purely artistic (geometric) perspective. When dealing with circle equations, one never perceives visually as one does when observing this drawing. As visual creatures, we understand the value of a picture, as it can convey what countless equations alone may not.

Uploaded

October 18th, 2008

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### Comments (21)

Prashant Rahenwal

Hi Jason, According to my darivation from the triangle side length the value of Pi = 180*x/asind(x), where x is the ratio of triangle side length to the diameter of cricle. From this we can say the even if we get limit of side of traingle upto microscopic level like u say plank length, but we can't have limit of macroscopic scale... The variable x defines the infiniteness of universe and we will not have value of Pi upto certain decimal places. It will not not stop until we get physical limits in both directions micro and macro.

L BARTEL PhD

By far the best explanation of Pi that I've ever read! To see it as a shape and use the equation myself nearly brought tears to my eyes. I will use this to teach many students. Keep up the amazing work!

Nancy Griswold

Really interesting, I read the caption too. Thanks for posting this in the group, I am not a mathemetician, but I do understand the concept about the triangles. Mathematical shapes are universally repetative throughout nature and art...this is all very interesting material you are presenting.