Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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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)
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.
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.
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)
This is extremely tricky to answer as ELI5. The answers you get with “circles are kinda everywhere,” are good, but I think they leave out a significant amount of the mystery.
The ones I saw trying to explain using Euler were not wrong, but quickly left ELI5 territory as well.
I think it’s worth acknowledging that the fact that Pi shows up quite as often as it does *is* surprising. Those of us in mathematics or related fields have gotten so used to the idea that we just take it as a given. That should not take anything away from the strange ability for Pi to show up in the absolutely weirdest places.
For instance, if we were to take the sum of the following numbers: 1/1 + 1/4 + 1/9 + 1/16…, we get the answer: π^(2)/6. I believe that any sane person looking at this should be wondering: where the hell did pi come from? If you are interested in this, you can look up the Basel Problem and find any number of wonderful rabbit holes to go down.
But that is beside the point for the moment. It is just plain *weird* that pi shows up there. Sure, once you start tearing the problem apart, you can see where it eventually creeps in, but I still think it is magical.
So what is the answer? I’m not sure there is one; at least, not one better than the answers saying that circles are everywhere.
It is legitimately strange that i ends up creating a relationship between π and e. It’s there. We can work it out. But it is not like we created i to do this. It just sort of happened. And I sometimes like to sit back and simply marvel at the fact that it did this.
My apologies for not really offering an answer as such. But I really want to emphasize just how wonderful and mysterious that π does end up everywhere, even in spots that would not seem to have anything to do with circles at all. It sometimes makes me wonder if we have mistakenly linked π to circles, simply because this was the first place we ran across it.
This is extremely tricky to answer as ELI5. The answers you get with “circles are kinda everywhere,” are good, but I think they leave out a significant amount of the mystery.
The ones I saw trying to explain using Euler were not wrong, but quickly left ELI5 territory as well.
I think it’s worth acknowledging that the fact that Pi shows up quite as often as it does *is* surprising. Those of us in mathematics or related fields have gotten so used to the idea that we just take it as a given. That should not take anything away from the strange ability for Pi to show up in the absolutely weirdest places.
For instance, if we were to take the sum of the following numbers: 1/1 + 1/4 + 1/9 + 1/16…, we get the answer: π^(2)/6. I believe that any sane person looking at this should be wondering: where the hell did pi come from? If you are interested in this, you can look up the Basel Problem and find any number of wonderful rabbit holes to go down.
But that is beside the point for the moment. It is just plain *weird* that pi shows up there. Sure, once you start tearing the problem apart, you can see where it eventually creeps in, but I still think it is magical.
So what is the answer? I’m not sure there is one; at least, not one better than the answers saying that circles are everywhere.
It is legitimately strange that i ends up creating a relationship between π and e. It’s there. We can work it out. But it is not like we created i to do this. It just sort of happened. And I sometimes like to sit back and simply marvel at the fact that it did this.
My apologies for not really offering an answer as such. But I really want to emphasize just how wonderful and mysterious that π does end up everywhere, even in spots that would not seem to have anything to do with circles at all. It sometimes makes me wonder if we have mistakenly linked π to circles, simply because this was the first place we ran across it.
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