Cat's Eyes Removed Ahead

I have written earlier about the strange fascination of British Signs and public postings... This one is like one I saw and mentioned in an earlier post.... Cats eyes are the British term for the little reflective things along the center of the highway.

Patterns in Iso-perimetric Problems - 3

In my last post on iso-perimeteric problems I finished with the question "If you have 360 feet of fencing and want to build a pen along a river or barn, the rectangular pen above is NOT the most efficient way to use three straight sections... assume you are limited to four fence posts, so you put two somewhere on the river, and can put two more along the fencing to hold straight sections of fence; how do you place them?

Now extend that problem to four sections of fence (and five fence posts)... and then generalize the results... "

The sweet mathematical idea to make all these types of problems easy is symmetry . Imagine that you had found the perfect solution to the fence problem with three sections and used it on your side of the river. Then the guy on the other side of the river decided to copy your master plan on his side...
NOW make the river really narrow..... no..more narrow... like a line.. Now erase the river completely...
What do you see when you look at your neighbors fence and your fence together? It must be a six sided figure.. and it must contain the most area it could contain for any six sided figure, since both sides hold the maximum possible... you couldn't make the total bigger without making one side or the other bigger...so you must have the most efficient six sided figure possible, a regular hexagon.
So what does your half look like? It must be half of a regular hexagon divided across two opposite vertices.
The solution would be to divide the 360 feet of fencing into three sections, and make the angles between the fence sections 120^o and slide the two ends up against the river.
To solve the three sided non-symmetric figure, we used 1/2 of a six sided symmetric figure. And the general solution from there, I hope, is trivial.

Actually with a little handwaving, this blog could have been much shorter..but I remember Pascal once wrote, "I have made this longer than usual, only because I have not had the time to make it shorter." [Lettres Provinciales, No. 16 (1657)]

A Variation on a Harmonic Theme

One of the Blog sites I drop in on regularly is Sol Lederman's Wild About Math. He has a contest there about every two weeks, alternating with another site so that they produce a problem each week. The latest one is about the probability of hearing all five songs in a set if they are played randomly and you listen for eight times.. I know a way to do it..but the contest is not quite over, so I won't give that away..

But it did remind me of a related problem... Suppose you decided to set it out and wait for the band to play all five tunes. I just heard a nice talk at Gresham College by the current Gresham Lecturer in Geometry, and head of the Cambridge Dept of Math and Theoretical Physics, John Barrow about just this type of problem. The answer, somewhat nicely, is related to the harmonic sequence I wrote about before , 1/1 + 1/2 + 1/3 + ... etc...

Here is how it works... The average number of times it takes to succeed at something that has a probability of P, is 1/P... for example, the probability of rolling a one on a regular die is 1/6... if you set down repeatedly and counted how many times it would take until you rolled a one, on average it would take six trys... sometimes you might get it on the first roll, and sometimes you might have to roll 10 or 20 times...but on average, it would take six rolls...
Now lets look at the music problem... with five songs, you are going to hear one on the first try...so that is pretty easy... but what is the probablity that the next song is new?
Well, if you have only heard one song, then 4/5 of the songs will be new to you, so it will take 5/4 or an average of 1.25 songs until you hear a second song... so for two songs, on average it will take 1 + 1.25 = 2.25 songs to hear both of them...
The probability of a third new song is 3/5, so it should take another 5/3 songs to hear a new one.. so for three songs we would have to listen to 1 + 1.25 + 1.667 = 3.917 songs on average...
If we write these out another way, we notice a pattern... the first took 5/5, the second took 5/4, the third took 5/3.. so for all five songs it would take 5/5 + 5/4 + 5/3 + 5/2 + 5/1 which is 137 /12 or about 11.4 songs to hear them all... and factoring out a five that is 5 (1/1 + 1/2 + 1/3 + 1/4 + 1/5)... the harmonic sequence..

Ok, how does that help.. well for five songs, not much... but what if there are 100 songs in your MP3 player... or 1000 song on your computer ... and you wonder... Hmmm how long would it take on a random shuffle to hear ALL the songs on my MP3 Player... well, 100 (1/1 + 1/2+ ... + 1/100) and that might take a little time to add up ... except for the incredible Euler... What Euler did was come up with a really good approximation for 1/1 + 1/2 + 1/3 + ..... + 1/n for any n... It turns out that as n gets bigger and bigger, the sum gets closer and closer to ln(n).. and Euler came up with a really good estimate of how wrong it would be. Today we often call the number gamma, or Euler's constant, but it is about .577... so if you want to know how long it will take to hear all 100 songs, just multiply 100 times (ln(100) + .577) .... I got about 518...
Ok, quick, let's check that with the answer we got for five songs.. 5 (ln (5) + .577) = 5 (1.609+.577) = 5(2.186) which is about 10.932....... compared to the actual 11.416.... NOT BAD for such a small number..

so for a thousand songs???? Well we leave that as a problem for the reader... good luck

Rheticus, and The Names of Trigonometric Ratios

Spending lots of time lately reading old English journal articles (1825-45) sent me by Dave Renfro who trys to help me stay up on the history of math. It is kind of great reading and watching the actual history of ideas unfold as they did in the old journals.... I came across an interesting letter from Agustus De Morgan about the protege of Copernicus, George Joachim of Rhaetia, also called Rheticus. It was Rheticus who managed to convince Copernicus to publish his long withheld manuscript.
I didn't realize for some years of teaching that in the early days the trigonometric functions were conceived to be lengths of segments in a circle of a given diameter (or radius) rather than the more modern view of ratios. I did not know until I read this article, that apparently it was Rheticus who first developed this approach. In fact, the tables he created to include in his publication of the trigonometric sections of De Revolutionibus were the first tables to include cosines (although he did not use these names). Here is the way De Morgan wrote it:
" Modern teachers (he writes in 1845) of trigonometry have pretty generally abandoned the system of independent lines, which used to be called sines, tangents, &c.; and have substituted, for the meaning of these words, the ratio of the sides of right-angled triangles. It appears that they have antiquity in their favor; indeed so completely has the idea of representing the ratios of the sides of triangles taken possession of the mind of Rheticus, that he abandons the use of the word sine. He dwells on the importance of the right-angled triangles, without any reference to the circle: his maxim expressed in the dialogue, is Triquetrum in planicie cum angulo recto, est magister Mathesos . It would also seem as if his choice of the semi-quadrantal arrangement with double descriptions was dictated merely by the convenience of heading one division with majus latus, and the other with minus latus. [Rheticus had labeled the top of his table with perpendiculum and basis, then the bottoms of these columns were reversed, much as Sine and Cosine were reversed at the top and bottom of tables used in my youth before calculators]........ The names cosine, cotangent, and cosecant are the consequence, not the cause, of this duplicate system of arrangements.......The introduction of the terms sine of the complement, complemental sine, and cosine, &c., followed after an interval of more than half a century."
De Morgan points out that one of the reasons it is so hard to find copies of much of Rheticus' work is that ", In the Index Expurgatorius, it is not Copernicus who is forbidden to be read generally; the prohibition only extends to the work De Revolutionibus, and is accompanied with a nisi corrigatur. But Rheticus is wholly forbidden to be read in any of his works. "
I think the difference in the two mens treatment in the Index may be because of the fact that Rheticus was Protestant, and in fact, was at Wittenburg, the very University where Luther had taught, and burned the Papal Bull.

An interesting anecdote told about Rheticus while he was, "puzzling himself about the motion of Mars, he invoked his genius or guardian angel to help him out of the difficulty: the angel accordingly lifted him up by the hair of his head to the roof and threw him down upon the pavement saying with a bitter laugh, 'That's the way Mars moves.' "

addendum James asked about the phrase "semi-quadrantal"... this just means he only went from 0 degrees to 45 degrees (1/2 of a quadrant) and then put Sin-Cos (he didn't use these words) at the top of the columns and Cos-Sin in the reverse order at the bottom... so that from 45 to 90 degrees was simply read up from the bottom....My old CRC tables were arranged the same way, and many textbooks did as well in the Fifties-sixties
 Giving Sin and Cosine as ratios was still pretty new when this was written by De Morgan. It appears that Peacock had initiated the practice in his lectures at Cambridge around 1830, and by 1837, according to De Morgan, it had become the accepted way to define the terms.