Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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If you wanna describe the universe around you, you need to build up from the simple things to the more complicated.
Start with points, then connect them with lines. You can measure lines to get lengths and compare them to get Ratios, Fractions, etc.
When you go to 2D shapes, triangles can be used to help describe any polygon, which is why trigonometry is so important (literally Triangle Measure).
But since circles aren’t a polygon (any shape with straight sides), you need to describe them differently. Whilst the most important feature of a circle is its radius, it’s easier to measure a circle’s diameter by hand. Because of this, historically, mathematicians have typically used a circle’s diameter as a reference point.
When this diameter is compared to the circumference (also easily measured physically), you always find the circumference as being 3.14159… times bigger than the diameter. With such an important ratio being so important to circles, they instead called it Pi for accuracy.
Bringing this full circle (pun very much intended) to your original question, “Why does Pi keep popping up in maths?” This model of the universe we’re building up starts with lines that give us basic numeracy, then build to Triangles and Circles. As you go further in maths, you keep using your previous work so as to keep it consistent, and so circles and Pi end up being used all the time, even if not directly in a circle.
TL;DR – Points are basic, Lines are useful, Triangles and Circles are extremely useful. You need Pi to describe Circles.
Source: A very nerdy maths teacher (me)
If you wanna describe the universe around you, you need to build up from the simple things to the more complicated.
Start with points, then connect them with lines. You can measure lines to get lengths and compare them to get Ratios, Fractions, etc.
When you go to 2D shapes, triangles can be used to help describe any polygon, which is why trigonometry is so important (literally Triangle Measure).
But since circles aren’t a polygon (any shape with straight sides), you need to describe them differently. Whilst the most important feature of a circle is its radius, it’s easier to measure a circle’s diameter by hand. Because of this, historically, mathematicians have typically used a circle’s diameter as a reference point.
When this diameter is compared to the circumference (also easily measured physically), you always find the circumference as being 3.14159… times bigger than the diameter. With such an important ratio being so important to circles, they instead called it Pi for accuracy.
Bringing this full circle (pun very much intended) to your original question, “Why does Pi keep popping up in maths?” This model of the universe we’re building up starts with lines that give us basic numeracy, then build to Triangles and Circles. As you go further in maths, you keep using your previous work so as to keep it consistent, and so circles and Pi end up being used all the time, even if not directly in a circle.
TL;DR – Points are basic, Lines are useful, Triangles and Circles are extremely useful. You need Pi to describe Circles.
Source: A very nerdy maths teacher (me)
If you measure from the center of a circle the distance to the edge will always be the same.
Since the distance will always be the same (definition of a circle)… If you make a triangle from the middle of a circle to a place around the circle the long end will always also be a consistent length.
Those triangles are the basis of trigonometry and geometry.
Triangles are really easy to work with. You can measure their sides and you can easily measure their surface area. Imagine measuring the area or volume of a box. Easy. Triangle, nearly as easy. Now imagine measuring the area of a jelly bean! That’s super hard. However, if we can convert a jellybean paper cutout to just a bunch of triangles… Easy again!
Even pi can be calculated that way. We break a circle into lots of pizza slices and then we know the area of each pizza slice triangle. When we add together a whole lot of really thin slices of pizzas we can approximate pi.
But once we have a sufficiently accurate measurement of pi, we can use that number in all kinds of triangles and angles.
If you measure from the center of a circle the distance to the edge will always be the same.
Since the distance will always be the same (definition of a circle)… If you make a triangle from the middle of a circle to a place around the circle the long end will always also be a consistent length.
Those triangles are the basis of trigonometry and geometry.
Triangles are really easy to work with. You can measure their sides and you can easily measure their surface area. Imagine measuring the area or volume of a box. Easy. Triangle, nearly as easy. Now imagine measuring the area of a jelly bean! That’s super hard. However, if we can convert a jellybean paper cutout to just a bunch of triangles… Easy again!
Even pi can be calculated that way. We break a circle into lots of pizza slices and then we know the area of each pizza slice triangle. When we add together a whole lot of really thin slices of pizzas we can approximate pi.
But once we have a sufficiently accurate measurement of pi, we can use that number in all kinds of triangles and angles.
It’s because circles are kinda everywhere. If anything is circular, cyclical or angular, it will include pi somewhere.
Circles are also really special, as they have the smallest perimeter to area ratio of any other shape. So it’s not random shape favoritism. They’re the most efficient shape in terms of that metric.
It’s because circles are kinda everywhere. If anything is circular, cyclical or angular, it will include pi somewhere.
Circles are also really special, as they have the smallest perimeter to area ratio of any other shape. So it’s not random shape favoritism. They’re the most efficient shape in terms of that metric.
I think pi is kind of fundamentally related to the “upscaling” of something from a lower dimensionality to a higher dimensionality (1D to 2D, 2D to 3D etc)
I think pi is kind of fundamentally related to the “upscaling” of something from a lower dimensionality to a higher dimensionality (1D to 2D, 2D to 3D etc)
Pi isn’t to do with circles. It is a fundamental constant underpinning many aspects of mathematics, one of those happens to be circles.
It is actually the period of a function that is the very core to many aspects of modern mathematics, but sadly I do not understand the field well enough to give an ELI5.
Pi isn’t to do with circles. It is a fundamental constant underpinning many aspects of mathematics, one of those happens to be circles.
It is actually the period of a function that is the very core to many aspects of modern mathematics, but sadly I do not understand the field well enough to give an ELI5.
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