Transcript

Where's the Math? | Ivars Peterson | TEDxACU

I've been taking pictures of mathematical patterns for many many years some of the earliest images that I have date back to 1967 I had just graduated from high school and I had a chance that summer to go to a World's Fair in Montreal the fair was full of fascinating architecture and here is one example this is you're inside a building looking up a spire right in the center of the building now if you look at that how is that spire put together well the basic unit is a hexagon six sides of equal length and it's arranged so this thing is a stack of hexagons it's then and it's arranged so that the corners of the upper hexagon fit on the sides of the lower hexagon very simple idea but a beautiful structure yeah here's another structure in the background there is the u.s. pavilion a gigantic geodesic dome now when you get close to it or inside it you see hexagons lots and lots of hexagons but when you think about it you know they can't all be hexagons why not hexagons fit together to cover a flat surface and this is the floor pattern and what in subway stations in Washington DC so how do you turn a pattern like that into a curved one but one of the possible strategies is to introduce Eunice with fewer sides and that's what happens in this particular playground climber is mainly hexagons but you notice there are some Pentagon's five sided figures introduced and it turns out if that were a complete sphere all the way around you would need exactly 12 Pentagon's to accomplish the arrangement all right now back to the u.s. pavilion in the background there somewhere in there maybe there should be some Pentagon's and if you look closely enough they're there they're hard to see because if this were a complete sphere you would need only 12 Pentagon's for that whole pattern there all right another structure this one is not at the fair is in Washington DC it's a sculpture it rises 50 feet into the air and it's made up of rods and cables all right now if you look closely you'll see that the rods are not connected to each other not directly so how can it possibly stand out right this is the base it's sitting on three legs and basically what's happening this is the cleverness of the design is that the cables are pressing on the ends of the of the rods and the cables in turn are being stretched by the rods and you have an interplay between those forces of compression and a tension that gives you allows this structure to stand up now if you want a clue as to what how you can get that into play to work you can step underneath and look up and you see this beautiful symmetric pattern but tells you something about the balancing required to get that structure to work now across the mall in Washington DC is another interesting building this is the East Wing of the National Gallery of Art it has a floor plan that is basically a trapezoid that has been sliced into two triangles now one result of that design is that you have a lot of corners that are not ninety degrees and that's unusual most of the time most of the places we go we see right angles everywhere but not here so now if you take a close closer look at one of the faces this is the western face of the National Gallery of our East Building if you look to the far right you will see a particularly sharp edge 19 degrees you don't see that every day if you look closely at that edge you'll see a dark smudge and this is you can see in fact this is a close-up now of that dark smudge what do you think that is any ideas well so many people have been so fascinated by that unusual edge that they've had to touch it and so that the marble has been stained by all those people touching it you go up to it about shoulder height typically and you put your hand there now this is more than just touching it it's a record of all the different people who touched it so it's a heightened distribution so this is an effect kind of population distribution built into that one marking now if you look really closely you will see up at the very top near the top there's another slightly dark smudge you think that might be well those are the show-offs I would guess who wanted to reach as high as they possibly could but there are marks of human use and human activity all over the place and you can learn from those a great deal about what's going on this is one example this is the door to the men's restroom at East Tennessee State University notice the paint has worn off up in that area well that also is men pushing the door opens the swing door so they push on the door and they leave their marks and again men are different heights so you have an up and down distribution and left and right that depends on how hard you want to push the closer you are to the edge the easier it is to push but you don't want to be right on the edge so you have that pattern all right now notice that the metal plate is lowered down why is that well the other side of the door has a handle which you pull and most people will pull at about waist height and will push at about shoulder height now what do you think the pattern would look like on the women's restroom just down the hall you picture it all right there it is it's lower down so the average height of women is less and it actually isn't quite it's closer to the edge to where it's a bit easier to push all right we see these kinds of wear patterns all over the place here's another example these are elevator buttons notice they've worn out well what can you say about what floor this might be on how people were using it and various are the kinds of questions you might be able to ask and then answer from that pattern here's another one this is the steps and Wells Cathedral in England which has been there for centuries but notice the foot traffic has worn the stone out in certain area so what does that tell you about the way people use this area and there are other kinds of distributions here's icicles hanging from a gutter well in this case it tells you something about perhaps water flow or the shape of a gutter or another kind of distribution rather interesting one these are Oregon pipes in a church the left side looks like it's a mirror image of the right side but why can't it be an exact mirror image well organ pipes the sound of our kite depends on its length so each note has its own length of pipe and so you could arrange them from smallest to largest or you could fool the eye and put them in this arrangement so it looks almost mirror symmetric but it isn't exactly there are all kinds of things to see as you go around next I have a sign but I found on the campus of Westminster College in Salt Lake City it's a campus organization of some sort but the symbol on top there is rather unusual is it's an actually a mathematical object called a mobius strip you can make one easily yourself by turning one turn and then attaching it to the other end so you end up with this loop it has a lot of interesting properties if you were to draw a line down the middle starting anywhere you can go around and around and around until you came back to where you started and if it in fact it has just one side you do the same thing with the edge - well this object you can see the same effect in a display at the Boston Museum of Science they put a track down the middle of this strip and they have a narrow train running around it continuously in the exhibit now people have been fascinated by this all sorts of people architects engineers mathematicians and in the next slide an artist decided to carve one out of granite big piece this is at the Baltimore Museum of Art now all of us came out of mathematical research about 150 years ago when mathematicians were interested in the question of how many and what kinds of different surfaces are there and wanted to figure out what all the different possibilities were but there is a place where you see this almost every day maybe several times a day you find it in the recycling 'some those three arrows up there if you join them together you'll get exactly the shape and that's deliberate the designer of the recycling symbol in 1970 wanted to suggest continuity and a finite entity in his words and as a result he came up with that design so you see it all over the place if you take a closer look at it there are the three arrows now look closely at the arrows there are three of them two of them are the same one of them is different and that reflects the fact you put twist into this so the Mobius strip is there but if you start wandering around you find that there's another version what's different about this one well the three arrows are identical that is the result of someone not paying attention someone saw three bent arrows and made three bent arrows without thinking about what it was actually showing now what is this object itself well it turns out that in itself is like a mobius strip and has one side but it's made by twisting this three times through that 180 degrees and when you join the arrows you get a knot instead of a regular form and in fact the recycling symbol for Santa Clara in California actually shows that mutant form but in its full form in this way alright another everyday object that you might even take for granted fire hydrant what can I possibly say about a fire hydrant well if you look closely at the top of it and at the valve used to turn on the water there's something odd about it it's a pentagon and it's five signs why well if you look at a standard wrench its jaws are parallel it means they work best when the knot or the knob has an even number of sides four six eight well in this case it doesn't work so well on a pentagon you need a special tool in fact to turn on the water like that now why is that done because you don't want just anyone turning on the water and a fire hydrant so I've just been built into the way in which people take care of this and I'm gonna finish with breakfast how many of you here eat Cheerios or some other ring-shaped cereal for breakfast okay fair number as you get done the Rings float as you get down towards the end you've been watching closely you'll notice that they tend to clump together or stick to the sides always why is that well there's a force called surface tension the liquid rides up each of the Rings and the sides of the bowl and when two rings get close enough together they attract each other so you have this automatic attraction that brings them all together into those clumps so next time you have breakfast look for if you have enough of these rings together you see patterns even more clearly notice that this is a organized itself all by itself into orderly rows offset rows and there's another pattern here too if you look at each ring is surrounded by six other rings so you have a hexagonal pattern you get something similar when you pile stack oranges or apples into a pyramid there you see the pyramid form and the offset rose but you also see again at an angle that same pattern of hexagons one Apple in the middle in this case is six around it all right this is just a very small sampling of the many ways in which math is embedded in the world around us at the same time when you look at anything very closely and carefully questions are sure to come to your mind and math is a wonderful tool for helping you understand what you're seeing so where is the math everywhere when you start looking you