Round (Circles and Spheres)


Merrill what do you think is cool about spheres in circles and all that from top logical standpoint spheres are pretty much the simplest shape you can think of. It's like join everything anything to a spear and it's still the same shape like take a taurus. Join us to it. just glues. fear. It's the same thing. It's kind of like the identity of top logical objects. Explain what you mean by joining us via taurus. So suppose he had just a tourist like a regular old donut and let's say you cut a hole out of that. Any cut a whole lot of the sphere and you glued those two together. The sphere would just kind of disappear into that torres just still be the same donut would still have the same donut hole. So tom logically. They're interesting interesting for other way for other reasons. I mean i like. I like the fact that they could be extended to any Dimensioned even though they're continuous objects could do that. With hyper hyper parabolas and pre aids. And all that stuff. But we're doing an episode on spheres. Just because they're cool they're the simplest object but we have all this math related to them. I mean you played as we're going to see to find his entire system of math using spheres and straight edges and so without further ado regal history of sears in circles which is a very difficult thing to do because planets are spherical. I mean spirited just exist naturally but i guessed We could talk about the early history. The early history. I mean people didn't understand spheres very well. They interested in a long time the study of pie until about a thousand see people that was between three and three point. One two five which is not very accurate finally euclid started using spheres and circles to do different proofs and he did this with compass straight edge geometry and merrill say anything about straight as geometry like doing explain maybe the limits of what you can do with head. I mean it's proved by construction. I mean as just like you can drop circles and he can drop fears and you can like get things such as length and circumference through them and you. Can you know do division to find out. it's like what ratios are things like pie. What's interesting though. Is ed using a compass straight edge. You can't do a lot of elementary things like You can't make us fear of twice volume of another sphere or you can't divide an angle by three or even five. There's a there's a very limited of stuff they can do with it however for a about a thousand years it was like the golden standard for math and geometry was euclid elements which we've talked about many many times on the show the proof that he had For the area of a circle being half the area of a rectangle is that divided the circle into a bunch of pie basically and made the slices of pie instantly thin and noted that those are triangles about the size of the radius tall with the total base length of the circumference of the circle. Right with you. Basically proved that a circle has area of a triangle with a base of the circumference and height of so. You could imagine this as listen. You take an orange right right and you peel apart the orange slices down so that all the points of the slices are facing up other orange skin. Really give you did that. In the in the orange an infinite number of segments you could make those segments simple triangles right. Yeah and then they together would all have the the area of the circle. Okay so pretty much you taking a bunch of triangles and adding them up to the area. The circle is what you're saying. Yeah and so the circum- the basis of all the struggles all put together. is equal to the of the circle and the height of all those triangles. Because they're all segments of the circle is our so the radius the radius and so one have ten circumference which is two times are leads us to the very familiar formula pyre squared or one half tower squared right. Because you know those of us who follow by heart will refuse to accept pi and tau which is to pie is the real circle constant or and then we go onto archimedes who proved that the volume of sphere is a two-thirds volume of the including cylinder and We did this problem episode or something but simply done what d- simply put. He did this by taking cylinder that encloses a hemisphere right and then a Cohen that starts at the top of the cylinder and goes down to the bottom and they noted that every with every slice the slice of the cylinder mine is a slice of the code was equal to the slice of the sphere so since the volume of cone is one third the volume of a cylinder. It's one of mine is one thirty goals two-thirds again. We might do this on a problem. Episode

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