So in the Greek times people were very interested in regular polygons, and in geometry. It was the dawn of geometry, really And they wanted to construct as many polygons as they could But regular, all the sides are the same length all the angles the same They where very interested in constructions with ruler and compass alone. Here's a good compass Ruler meant just a straight edge No markings on it at all. This one has markings but I'm gonna use it anyway. I won't use the markings. Here's a nice circle, drawn with a compass, in the old style So if I pick a point in the circle, I'm gonna draw a hexagon In a way that everybody will know, who has ever played with ruler and compass I'm keeping the compass length the same as it was for the original circle. Oops, ran out of ink there There we are.
But the ink is so satisfying to me that I don't mind the splodges occasionally All the circles pass very nicely trough the center, showing that I haven't changed the width of the compass And now I have six marked points on the circumference of the circle And if I want to I can put lines through them and I'll get a beautiful regular hexagon Make the lines longer looks even nicer Ruler and compass constructions are something else. There we are, a beautiful hexagon. And of course if I just connected three of the points, like these three, I would get a regular triangle. If I have any line, I can divide it in half with ruler and compass alone. Think that's worth doing just to show
how it's done. I'm gonna use it a lot later. So suppose I have line and suppose I have a couple points that were already marked on it and I want to construct the midpoint of
that segment and a perpendicular line through the mid point.
I can do that with my ruler and compass And I make some pieces of a circle on each side and I look at the places where this circle meets them and then if I connect those two points
those two new marked points I get a perfect perpendicular bisector of this segment. So the the ancients knew how to do a triangle and a hexagon the way I did it. They knew how to do a square 'cos you can make these perpendicular lines and that makes it easy to do squares. They knew how to do a pentagon. That's a little more complicated and that's all they knew. They could put
three and five together to get a fifteen-gon too but for instance you know what's the
next number we know how to do 3, 4, 5 6, 7 right, can you do seven? No one could say. And they asked can we do this? Is it possible to construct a 7-gon or a 9-gon, or anything else anything else at all other than the ones you can get by taking
three or five or three times five, or dividing those things by two you can divide again
and again by two just the way I did here to get like a 30-gon or a 60-gon that's all easy once you have a 15-gon. It was an
open problem from antiquity for two thousand years people played
with the ruler and compass and worried about that and then something
amazing happened it got connected with the rest of mathematics that often happens in mathematics when, when a problem gets connected to something else it becomes
accessible actually this fell to a prodigy, a pimply
19-year-old Carl Friedrich Gauss in 1796 what he did initially was just to prove
that you could construct regular 17-gon with ruler and compass and later on he'd analyze the problem
completely and and said exactly what you can construct
I'm gonna do the construction for you that Gauss proved it was possible but if I really
do it with a ruler and compass then you'll be very disappointed because
it won't work and that's just because the little
errors that I make along the way, there are are so many little constructions to make
they're all the same and they're all really easy but by the
time you're finished the errors have all built up and so I might construct a 15-gon [instead of a] 17-gon there's my circle so I wanna ... I would like to
construct a regular 17-gon in that circle so here goes the first thing I'm gonna do
is to draw a diameter. So the next step will be to
construct a perpendicular bisector of this diameter. Maybe I'll do one one such construction just so I'm not
cheating completely I know where one point is already the center, so I don't have to draw that then the other. I guess I want it to go on fro a while. Okay you know I can divide things in half with ruler and compass so I'm gonna divide them in half and I'm just going to do it by eyeball. We
could do it with ruler and compass but it gets tedious after a while. And then I want to divide each of these segments in half too That's a half, and maybe that's a half, like
that and then I get a draw another line do that with that the real ruler through
the quarter mark and then I'm going to do something
surprising I'm gonna draw a circle which has this radius this is half the length of the radius. This radius and based at that
point on this intersection this quarter mark (a little more ink in there) get it in the hole, and I'll draw this semi-cirle. Now I'm gonna draw a couple more lines. This construction is a lot of lines and it's really quite easy. Brady: Do you enjoy this?
Prof: I love it I love it I don't know why but its like origami or something it's very very precise craft stuff and it does something beautiful. Gonna draw that line I'm gonna draw this line. I'm gonna divide them both
in quarters too. So you know I can do that, I'm gonna do it by
eyeball again you know I could. So I want that quarter and I want this quarter and I want this one, there's a half, and this one Alright, and now I'm gonna draw some more lines
through this point of intersection and through
those two new points and I'm interested in where they
meet this line down here there's a fundamental point that's gonna be
useful to me and then another okay so now I have these two important
points on this line marked and I'm gonna draw I'm gonna do something funny, I'm gonna bisect
this segment from this point to the edge it's about
here but we'll hmm, maybe I'll do that accurately
because this is important and I'm gonna draw a circle whose centre's at that midpoint and whose radius is this it'll be inside the circle like that. I'm interested in this point, by the way, where my circle meets this line. And I'm gonna draw a tiny little circle around the center point with radius this
little distance here So now I have another circle and what I want is the line joins them there and now I'm gonna draw a circle which is centered at this funny point and this radius so one more circle we're
almost done almost there
Brady: you still enjoying it? Prof: I'm getting a little tired of it to tell you the truth
then draw that circle and the crucial thing is where it meets this line
there okay now we are ready you might say well where's the ... where's
the 17-gon and I have to draw out a perpendicular to this diameter which meets at that point and another one here
so I'm gonna draw a line here and a line here.
Should I eyeball it or should I do it for real? If I had little more energy at this point I'd use the compass, but what I'm
interested in anyway is this point and corresponding point here now you might if you're looking really hard
you might say that's not a 17th of the circle and in
fact it's 2 17ths of the circle Or it would be if I had done everything exactly right so I have one more division to do I draw this line I divide it in half and I get a point about here I would say that's the point and now I have my 17th of a circle, finally. By drawing this radius One radius, and two radius So this distance, from here to here, is supposed to be a 17th of the circle. Now just for fun and you may have to go off camera for this Brady let's see how how close I came
Brady: (laughing) alright Prof: now I would say I didn't get quite halfway between 'cos I estimated that too ... three ... ... nine ... ... ten ... there's another and another, well, gee, I'm it seems it's close to a multiple, and a last one. Brady: what did you make? What -gon have you made?
Prof: Let's see: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 16, 17, 18, 19, 20 21 so I made a 21-gon by accident an impossibility so it went wrong
because its I think probably not even a
numerically stable procedure even if I'd tried to use the compass at every time and done it as accurately as I could.
By the time I made all those divisions things little errors would have built up
and it would have come out wrong one way or another but um its provably the right thing if
you were exactly right let's see how'm I doing? so this is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15 Here's 16 and then since it's a 17-gon maybe it should connect this way Brady: (laughing)
Prof: not quite a regular 17-gon but what can you do? we used to have the address 1000
Centennial Drive because really we are actually a small
access road, Gauss way, and Centennial Drive is the big road that's
nearby and well was not a problem for us but it was
a big problem for our neighbors our neighbors are the Silver space lab and they build satellite experiments for NASA and so
these big big trucks would come rumbling along with some delivery or to pick up an experiment and there was
no 1000 Centennial Drive so they would rumble right on by and that was a continual source of trouble for them so they finally convinced the University
of California Berkeley which is the one who owns the land to
name the access road and they had some idea that I didn't
much like for naming it after a double star whose colors were blue and gold I forget the name the star actually and I
thought astronomers might know that but mathematicians won't know it won't mean
anything so I went to them it is said to my colleague maybe we should
name it after some famous scientist and Gauss would be perfect because he was both
a mathematician and he ran the ducal Observatory in the town where he was, so he would be
... he was very important in astronomy actually too and my colleague at the time said oh you
know you think he was an astronomer but the
astronomers will never heard of him and I went over to the director the
space lab and I said you know think we oughta name it after some famous
scientist, how about Gauss? he looked at me kinda surprised he said
oh, was Gauss a mathematician? so that was a done deal right away and then they were very generous they
said okay you you been nice to us you can have
number one and I said no we don't want number one
we want number seventeen because that represents Gauss's first
big splash as a mathematician and we very much about youth in mathematics and the success of post-docs and things like that cultivating young talent so we got to be
17 Gauss way and on our front doors instead a number we put the construction the 17-gon that I've just showed you.
The best part of this whole thing is him describing how his building got their address at the end.
This is a numberphile guy. All these old mathematicians have that Bob Ross way about them. If you want the afro you'll need to switch to chemistry videos with Martyn Poliakoff. He basically stole Ross's hairstyle.
Looks like he's trying to communicate with the heptapods
r/ASMR
I love that his ruler is labeled with his name.
I've always wondered, rather than for drawing shapes, what practical applications this has on the real world?
He sounds like Mr. Burns
Geometry with Bob Ross
TL;DW 10:30 he accidentally draws a 21-gon.