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.