Chain, Chain, Chain, ...Chain of squares...

or All roads lead to two or one... guess Rick Regan over at Exploring Binary will like this.. and I think the actual next line in that song was "chain of fools", (rock me, Aretha) and yet I continue.... If you take a number, square each of its digits and find the sum you will get a new number(usually, is there more than one number that produces itself as its own iterate under this process). Do the same to that number, and the sequence continues. But eventually you have to come back where you started. Numbers with more than three digits will always produce a smaller number and three digit numbers will always be less than 9²+9²+9²= 243, so eventually, wherever you start, you end up with a number less than 243 (and 243 --> 4+16+9 = 29 so it gets even smaller).. what would be the last number that produces a number larger than itself? Ok, simple stuff, if you start with one, you get one and that's dull so let's go on. If you begin with two you produce the sequence 2--> 4--> 16--> 37--> 58--> 89--> 145-->42-->20-->4 and then repeats the cycle of eight numbers forever. Now the big conjecture... start with ANY integer (oooohhhh, he said you could pick ANY integer, how bold) and eventually, either it gets to one, or it jumps into this chain. Now I wonder; is there a way to prove (short of hacking them all out, which I have done) that there is no "other" chain that some numbers might drop into? 

John Cook did a computer search of all the numbers less than 1000 and they all went to one of these two absorbing states.

 I also have a feeling that as the numbers go off to infinity, the proportion of numbers that go off to one has some non-zero limit; in fact, I suspect it might be around 1/7 or just a tiny bit more, but don't have a clue how to prove that. Any takers? All conjectural rationale will be considered. I think the same kind of questions could be asked about forming the sum of the cube of the digits... would all roads lead back to one or two then? A quick answer trying only a few numbers... NO, Follow the orbits of 2 or 3 or 7 and they all go to separate self-replicating numbers or one-cycles.. 4 goes into a triplet of 55 250,133,55; so I guess my new question about cubics is... How many of these finite cycles are there?

Sunday Morning Math and Coffee

Sunday morning in East Anglia, and I am sipping my coffee as I read through some of my favorite blog sites... Do get over to Concurrencies where Steve Phelps has put up his favorite math movie, a really great explanation of Mobius transformations that should be seen by more HS students.

And since I have several blogs about coincidences, I should point out that Jon Ingram, also here in England somewhere (sorry Jon), has a nice simple problem about coincidences as well at his "Lessons taught, Lessons learnt" . Here is a teaser to get you started, right from Jon's page...

The average of a set of 64 numbers is 64.
The average of the first 36 numbers is 36.
What is the average of the last 28 numbers?

Then he very nicely answers the question:

Solution

The total of all 64 numbers is 64 X 64 = 4096.
The total of the first 36 numbers is 36 X 36 = 1296.
The total of the last 28 is therefore 4096 - 1296 = 2800,
which makes their average ²⁸⁰⁰/₂₈ = 100.

and adds, "and is it a coincidence that 64 + 36 = 100?"

OH! and I couldn't remember this, or where I saw it, but I found it again on Jon's blog... did you know that "1% of a day plus 1% of an hour is exactly 15 minutes"??.
Time for a refill, and then to touch off the notes on my next post...

A Serendipitous Coincidence? The First-Ever Pursuit Problem.

Just reading through some old copies of the Mathematical Spectrum from my great source of mathematical periodicals, Dave Renfro. Intrigued by a couple of posts about a problem from the ancient Chinese Chiu Chang Suan Shu or Jiuzhang suanshu (Nine Chapters of Mathematical Art...about 150 BC) submitted by David Singmaster.

The problem: "A water weed grows 3 feet on the first day, and its growth on each succeeding day is half that on the preceding day. A reed grows 1 foot on the first day and its growth on each successive day is twice that of the preceding day? When are they of equal size?"

Go ahead, stop reading for a minute and try to solve it because I give an answer (actually two different ones) below and I don't want to spoil the fun.

The interesting thing to me, was the two letters of solution. My pre-calc students came through exponential growth and decay a few chapters ago, so they would approve of the first solution that was submitted. It suggested that we assume that the height of the water weed was growing according to the exponential function hw= 3 (1.5)^d-1. The reed would reach a height of hr=3^d-1. Setting these equal we would find the heights are equal when d= log₂6; or at about 2.585 days. They also pointed out that the mutual heights would be 5.705 feet.

Simple, quick, and "Wrong" according to the next commenter. They pointed out that a careful reading of the problem stated that the water weed "growth on any succeeding day is half that of the preceding day", would meant that it grew 3 feet the first day, and then successively it would grow 3/2, 3/4, 3/8 ... feet on each day.. to find the height we should sum this geometric series. So he suggests the height of the water weed would be hw=. This would result in the weed being somewhat shorter after each day than in the previous solution. In the same way, the reed should, on successive days, change by 1, 2, 4, 8 etc feet, so it would have a height of 2^d-1 after d days. Setting these equal we see that the two plants will both reach a height of five feet (exactly) after d= log₂6; or at about 2.585 days.???? WAIT, that sounds familiar....where have I ... Oh YEAH!!!, that was the answer to the problem done the first way? WOW, what a lucky coincidence........ Well, NO... the good Professor who posed the problem stepped up to assure us that, in fact, if you used those same two approaches to similar problems they would always reach equal heights at the same time... (can you prove that???)

Try for yourself. I assumed similar means that the shorter grows at r times its previous days amount, and the taller at 1/r times the previous day. Try a few. In fact, it is frequently the case that the second method gives an integer solution, and when it is not, it seems pretty easy to adjust the growth on the first day of the two plants to make it come out an integer. If we call the first day growths W and R, then the solution works out to log_r(rb/a); where r is the common ratio of the plant whose growth each day is increasing.

Dr Singmaster points out that this ancient text solves both quadratics and cubics, as well as systems of equations (including indeterminate systems with more unknowns than equations) using the "modern" method of elimination. It also is the earliest known text to have used negative numbers, and includes the rules for all the arithmetic operations. It is also the oldest know source of chase or pursuit problems, such as this one. This is problem 11 of the 7th chapter. I had the good fortune to sit with my beautiful Jeannie as the only two non-Chinese speakers at a discussion about the text at the Needham Research Institute in Cambridge by a group of English and Chinese experts. The amount I was able to take in is a credit to the patience of the gracious hosts.

An Average Dismissal? rant about education and bad statistics

It was on my I google news, I had to read it...
"(CNN) -- A school board in Rhode Island has voted to fire all teachers at a struggling high school, a dramatic and controversial plan aimed at shoring up education in a poverty-ridden school district.
On Tuesday night, the board approved the plan by Frances Gallo, superintendent at Central Falls School District, to discharge 88 teachers at Central Falls High School."

ALL the teachers, NOT A SINGLE ONE that was capable...Did they fire the principal that hired and managed 88 incompetent teachers? (bet they didn't)...

SO HOW BAD WERE THEY? you ask.

"The firings come over the district's concern that teachers refused to spend more time with students to improve test scores." AHA, Was the superintendent ready to PAY them to spend additional time... time outside their contracted 5, 6 or 7 hours a day of classroom instruction??? (oh, they didn't say anything about that.)... Try to imagine another profession where that kind of demand would even be suggested... Hmm, are all the bankers who got us into this economic mess actually putting in extra hours at lower pay to help us get out...(pssst.. the correct answer is No.)..

But a teachers' union spokesman called the firings "drastic."

As my students would say....DUH!!!

And now the part I love, they use statistics.... (ummm poorly)...
"The spokesman also cited a 21 percent rise in reading scores and a 3 percent hike in math scores in two years."
Ok, What does that mean? What scores went up 21%.. Try to guess. Does that mean the median national percentile for the school went up by 21 percentile points ? Were they at the 20th percentile and now they are at the 24th (a 21% increase) or did they perhaps go from the 20th percentile to the 41st percentile (an increase of 21 percentile points).. Does it mean that EVERY kid in the school improved 21% over his previous (raw score? natl percentile score ?) WHAT DID THAT MEAN???

And of course, the article concludes with a stunning judgment of the school and the community...."Central Falls is one of the lowest-performing schools in Rhode Island." (Wow, at some schools they probably just took the teachers out and shot them...)

The statistics students in my class have already learned to ask..."Measured HOW?" Do they have the smallest number of people in the theater program? (see, that is a pun on "lowest performing"... rim shot please.... and it might be the truth.. we really don't know)

I have no idea how good, or bad the teachers in Central Falls are, but I am willing to wage heavy odds that they almost certainly are not the 88 worst teachers in Rhode Island. In fact (no elementary or middle school teachers were dismissed???) they may not be the 88 worst teachers in Central Falls... Wait one more wager... I be the superintendent is about to become much more familiar with the ins and outs of class action law suits...
and amazingly, there will be lots of people lined up to interview to work there next year... Eddie Izzard does a nice routine on the Anglican church in which the punch line is "Death or cake?" and in this economy, there will be people lined up to say.. "Death, please."...And now Mr. Ballew has left the soap box...next speaker...

Polynomials, Coloring Graphs, and Sudoku

Did you ever get half way through a really tough sudoku and begin to wonder if it really has a solution, or finish one and wonder if when you decided that one cell was a five and not a three that if you had picked the other one it STILL would have worked out? The answer, my friend, lies in polynomials.

Recently learned about all this from a nice article you can read for yourself, and maybe if you find my mistakes, drop me a line.

The authors point out that every Sudoku puzzle can be thought of as a graph with 81 vertices (nine rows and nine columns). Now think of the nine numbers in the top row..they would all have to be connected to each other with an edge. They would all have to be connected to each of the cells that are in their same column, and also to the eight others in their sub-square (but they have already been connected to at least four of those, two in their row and two others in their column). By my calculation, that means that each of the 81 vertices would have 8+8+4=20 edges connected to it, for a total of 810 edges.

Now if we associate the number in each cell as a color, a solution would be the same as a properly colored graph (no edge connects two vertices of the same color). An unfinished Sudoku is then a partially colored graph, and solving it is just a matter of finding a way to color the remaining vertices so that no edge connects two vertices of the same color. If it is not possible to do that, there is no solution... and if you can do it more than one way, there is more than one solution.

Now comes the polynomial part. It seems it is "well known" that "The number of ways of coloring a graph G with n colors is well known to be a polynomial in n of degree equal to the number of vertices of G." (Where have I been?). Wait, that means we are looking for a polynomial with degree 81... Ummm.. that may take a little time, so let's look at a simple example.

Suppose we have a simple graph like a triangle, the complete graph of degree three, usually called K₃. [A complete graph has each vertex connected to every other vertex with an edge.] It should be clear it can not be properly colored with one or two colors, but with three colors we could do it in six ways (3!). But how many ways could you color if if you had four colors available, or five, or nine, or a complete box of 64 crayola crayons. The answer is pretty easy using the associated polynomial. For K₃ the number of ways to color it using n colors is n(n-1)(n-2). So for four colors you could complete the coloring in 24 different ways. You can easily extend the polynomial for a complete graph of t vertices using n colors to get K_t to be n(n-1)(n-2)....(n-t+1).

Ok, so what about one like this? Can you do it in three colors, four, five? If you can write the polynomial for it it should tell you. I admit that I am not sure about how to write some of the pretty simple ones... for instance, if a vertex is joined to two vertices that have n-2 colors possible each, but they are not connected; should that new vertex contribute a (n-3) or an (n-2). Trying to learn as I go along, but mostly by drawing simple ones that I can enumerate all the possible colorings.



Complications spring up when you have a graph that is more complex (and especially if it has 81 vertices), but if you had the polynomial, then you could just evaluate it for n=9 and it would tell you how many solutions there are. The authors did point out some simple theorems that pop out when you make the connection to graphing. For example, any sudoku that gives you only seven of the nine numbers in the "given" must have more than one solution (if it has any). Consider that if you have a solution you could interchange all the cells that have the two missing colors and get another solution. So a puzzle with a unique solution must contain at least eight of the nine numerals. The authors suggest that the number of cells that must be "given" in the problem is an unsolved problem, but they have unique examples with as few as 17, so that would be an upper bound on the solution. The puzzle at the top has eight of the nine digits in it, and exactly 17 cells "given". So does it have a solution? or two? Happy graph coloring children.

Here is a simpler problem, can you write the coloring polynomial for this graph, and find the number of colors for n=3, 4, and 10 colors?

Hallucinations, Polar Graphs, Alan Turing, Logarithms and the Leopard

So how does all that go together... it's all about brain chemistry. In a recent article in Plus Magazine (you ought to read it regularly), Marianne Freiberger, one of the Plus Editors gives a really nice capsule history and current status of the research going on about the brain and hallucinations. Amazingly it ties together the entire cast of characters above, and more.. Here is just enough to tease and entice you to go read (study?) the article.

It seems that the hallucinations induced by some drugs and certain illnesses fall into identifiable sets, four of them by one classification scheme, spirals, and funnels (these are often reported by LSD users); honeycombs (from marijuana most often) and different types of lattices like cobwebs or triangles. . Research about what happens in the brain that causes us to see hallucinations has also helped us understand the very mathematical model of how the brain images the world onto the cortex. Amazingly it seems that the imprint on V1 (the part of the cortex where vision is processed first) from the field of vision is achieved by translating a polar graph into Rectangular coordinates. (Do you see a pre-calc project building here?)

The transformation simply turns the polar (r,q) into (log(r), q). (this assumes r is sufficiently large, apparently there is massive complication if you let r get too small. ). So for example, as they point out in the article, a circle in the visual field will plot itself to a straight line in the rectangular coordinate field of V1 (I imagine something like a string of adjacent brain cells being activated). Conversely, a ray emanating from the center of the eye would trace a line perpendicular to that. Now just wait until the next kid asks me why we need to learn to use polar coordinates...B A M...

Now the hallucinations seem to start in the same V1 area of the cortex, so they have been investigating how the illness or drugs can cause the cortex to distort the signals it gets from the eye. It turns out that the theory most helpful, is the theory that Alan Turing, the computer guy, came up with to explain how animals got stripes and spots, like the leopard.. Turing used differential equations to model how two types of agents, an inhibitor and an activator could coexist in an animal and explain the stripes or spots in terms of the rate at which the two chemicals diffuse through the skin. They even have a interactive applet that illustrates the process (Plus is really good about that kind of stuff). Now Turing's model is just the stimulus for a much more complicated approach of the brain behavior of hallucinations, but still a nice story to share with kids.

Speaking of nice stories to share with kids; I close with a great story I heard at a lecture at the Center for Mathematical Sciences at Cambridge from a young lady (sorry I don't have my notes here) who was talking about the Enigma project in Bletchley Park. Alan Turing, of course, was instrumental in the code breaking efforts there and it was while he was there that he laid the foundation for the programmable computer. Now Bletchley was a VERY secret project, and it wasn't made public until years later. She told of a young man who spent the war years there without being able to tell anyone what he was doing. After the war one of his old teachers walked up to him and cursed him for having hid out at a desk job while his friends gave their lives in the war. Turing, it seems, became a problem for the British Security services when their fears that his homosexuality would make him subject to blackmail and therefor a security risk. He eventually was hounded out of public service and could not get work.

Now the side note is that Turing was fascinated with the story of Snow White, and when he was found dead there was an apple laced with arsenic (ok, found out it was cyanide, and apparently they found it in his body, but no one checked the apple??? on his night table beside him, with one bite out of it. There is some question about whether a tortured Turing killed himself, or if he was done away with by the paranoid security agencies. Whichever, the story was passed around by computer geeks down through the years. Then as two young computer nerds were developing a really cool new approach to computing, they decided that they would honor Turing's part in the computer process by symbolizing his death in their logo, an apple with a bite out of it. The story, as she said, is too good not to tell, even if it is totally untrue.

That Ain't Fair..... is it???

An Encore Blog

Had a chance to take ten of my students down to the Center for Mathematical Sciences in Cambridge one night a while back to hear a lecture by Richard Weber of Queen’s College. He is the one in the right on the picture during his appearance on the British TV game show, Who Wants to be a Millionaire. He also happens to be the Churchill Professor of Mathematics for Operational Research. His primary work is in problems in communications and systems, and the mathematics of optimization, algorithms, probability and game theory.

On this particular night, he was talking about “The Disputed Garment Problem”, an ancient problem involving the concept of fair division. It seems a simple idea. We all think we know when the division is un-fair, and if we picked an honest, unbiased third person to make the division, it would seem that a fair result would follow, and yet, it seldom does. If you ask people to describe the rules that a fair division would entail, most people can come up with several rules, what a mathematician would call the axiom set, and probably agree with each other. But then when you make a decision consistent with those rules, the same people will scratch their heads and say, “Wait, that’s not fair.” What is even more scary is that often they can make five or six rules, and you find out that it is Impossible to meet all those rules.

So let’s look at a simple example. The Babylonian Talmud is a compilation of the ancient laws from an oral tradition set down during the first five centuries after Christ. They serve as the basis of Jewish religious, criminal and civil law. In one of the problems the Talmud offers: Two people dispute the possession over a garment. One claims that he should get half, the other claims all of it. The solution of the Talmud is that the first gets ¼, and the other gets ¾ .

The Talmud is a terse document, (sort of like 19th century math books). It tells you what to do, but not why. In this case however, the reasoning seems somewhat clear. Since half the garment is not in dispute, it should go to the one who claims the whole. The other half, claimed by both, shall be shared equally, giving the 3:1 split.

Given that beginning, a more interesting problem involves sharing by three wives in what is called the marriage contract problem: A man has three wives whose marriage contracts specify that in the case of his death they receive 100, 200 and 300 units respectively. But when he dies he leaves less than a sufficient amount to cover the 600 debt. How should the estate be divided.

The answers the Talmud gives for different amounts is surprising, and somewhat confusing. If the man leaves an estate of 100 units, the three wives will divide it equally (see below). If the estate has a value of 300 the wives divide it proportional to their claim, each getting ½ the amount they were promised. But what is happening in the case with 200 remaining? The question for the reader is, what will be the division of an estate with 400 remaining? (I will give the answer later, but don’t cheat, try to work it out. And additional clues will come if you are really stuck…. I was)

Estate..wife 1...wife 2....wife 3
100.....33 1/3...33 1/3....33 1/3
200.....50 ......75........75
300.....50 ......100......150

Ok, so which of those seem like a fair division to you, and which do you think most people would agree with. With your answer tucked away, here's a similar problem with a different twist. Three people want to build a runway for their private planes. For Pilot A, the cost of a runway for his plane would be 100 units. For Pilot B, with a bigger plane and therefore a longer runway needed, the cost would be 200 units. And Pilot C, needing an even longer runway, would have to pay 300 units. If they form a coalition, they can build one runway big enough for all three for the 300 units. How should the share the costs?

Should they pay 100 units each? Probably not, the first guy would see no advantage to the coalition. What about each paying half of what they would pay for their separate runways? If that sounds fair, then look again at the marriage contract problem above for an estate of 300; are these really the same problem? The numbers are all the same. The only difference is that in one case they are sharing a debt of 600 units and paying a combined 300, and in the earlier case they had a combined credit of 600 and sharing a combined payout of 300. Is fair in one case the same as fair in the other case?

The good professor provided another alternative to the airplane problem, using a method called the Shapley value, named for Lloyd Stowell Shapley, from UCLA. The method is founded on objection and counter-objection of each party being balanced. But an easier way to understand the value is to take every possible permutation (different orders of the pilots) and how much each would pay if they joined the coalition in each order. For instance if they joined in the order 3,2,1, the large plane pilot would have to pay for the whole runway, the other two would pay nothing. Here is a list of ALL the possible orders and the amount each pays:

Order.......Pilot 1...Pilot 2...Pilot 3
1,2,3.......100.......100.......100
1,3,2.......100 .......0 .......200
2,1,3 ....... 0.......200.......100
2,3,1 ....... 0.......200.......100
3,1,2 ....... 0.......0.......300
3,2,1....... 0 ......0.......300

Now if you add up all the possible amounts, the six ways add up to 1800 dollars. Since pilot one pays 200 of the 1800, his share of 300 should be 1/9 of the 300 units, or 33.33 units. Pilot 2 should bay 5/18 of the 300 or 83.33 units, and the last pilot should pay 11/18 of the 300, or 183.33 units. Now that’s what I'd like for public service, good progressive taxation… make the rich guys carry the burden.

I expressed my dislike for the three wives solution on the night. I explained that if my bank went bankrupt and offered me 10 cents on the dollar, and paid another guy 50 cents on the dollar because he had more money, I would be screaming like a stuck pig. But if fair on one problem is fair on the other, then I’m REALLY not liking the rich guy getting 61 percent of his money back, and the little guy gets 33 % return.

Ok, so I’m going to make it a little more complicated in a minute, but I want to go back to the case of the three wives in the Talmud. If the rule is consistent, then any two wives should get the same amount they would get if there were only two of them and the estate was the sum of what they shared and the division was by the disputed garment solution. Ummm, let me try again. In the case for 300 in the estate, wife 1 and wife 2 get a total of 150 units, and their claim is for 100 and 200. If we use the logic of the disputed garment, then since wife 1 only claims 100 units, wife 2 should get the 50 extra right off, and they should share the remaining 100 equally giving a solution for the two of 50, 100… ok, that works, and if you check the other two possible groupings (wives 1 and 3 and wives 2 and 3) it also works out. So maybe this is the Talmudic approach. Can you find the solution for 400 units in the estate now?

For over 1500 years scholars studied the Talmud and couldn’t find a rule to explain the determination of values that was more specific than the general rule I just gave. Then Robert Aumann, a professor at the Hebrew University of Jerusalem, showed that the solutions were the ones you would get by using the Nucleolus for the coalitional game (I will spare the general reader the several pages of equations with subscripts and other assorted devices of mathematical torture that explain Nucleolus, but if you are, like me, one who shouts “Show me the Math”, you may find an extended description here. .)

Now, if you STILL haven’t figured out how much each person gets with the division of an estate of 400 units, here is a physical method that, remarkably, gives the solution for any amount less than the 600 units promised the three wives.




Imagine the image shows six interconnected tanks. The two tanks on the left hold fifty units of liquid each and represent the share of wife 1. The second set has two tanks of 100 units each, and represent the second wife's promised 200 units, and the third set of two each hold 150 units, to represent the 300 units of wife three. Now we pour an amount of liquid equal to the value of the estate into the top three tanks, divided equally between them. Behold, what is left when the connection across the bottom is allowed to do its thing, is the Talmudic solution. The solution above shows the answer for the division of an estate of 100 units. Here is the one for 300 units to help you see the physical solution method.

Now admit it, even if your not a math/science person, even if you don’t think the solution is fair, that’s a pretty cute trick… ahh go on, admit it.

Ok, if you STILL don't have the answer for 400, the division is 50, 125, 225....

Math and Predicting Earthquakes.

I was just learning about the Gutenberg-Richter law. I always knew that the intensity of earthquakes followed a logarithmic scale, that an earthquake of magnitude 4 would be ten times as powerful as a magnitude three earthquake etc. But I had never known that the frequency of earthquakes follows such a scale.

The Gutenberg-Richter law says that in any general seismic zone (not just a single fault area) the number, N, of earthquakes greater than some given magnitude, M, will follow the equation log(N)= a-bM. So if b is about 1, and it seems from one paper that is pretty close, then the number of magnitude 6 earthquakes will be only 1/10 of the number of Magnitude 5 Earthquakes; and we should expect 1000 times as many of magnitude 3 than there were of magnitude 6. In short, lots of little earthquakes and not many big ones. It turns out, reading a couple of research papers from people who study this stuff, that the Southern California area has a b value of one.. how convenient.

One problem with the law is that you need, it seems, really accurate measures of a LOT of earthquakes. Here is how one professional wrote about it, "Seismic hazard analysis is very sensitive to the b value (slope). If earthquake rates are based on the number of M ≥ 4 earthquakes, for example, a b value error on the order of 0.05 will cause the number of M ≥ 7 earthquakes forecast to be wrong by 40%. The value of b and its error is often miscalculated in practice, however. The common technique of solving for b with a least squares fit to the logarithm of the data, for example, leads to an answer that is biased for small data sets and apparent errors that are much smaller than the real ones."

The preferred method uses Monte Carlo simulation, but to reach the 98% confidence level on a slope accurate to that .05 error requires about 2000 earthquake records.

In thinking about this, it seems earthquakes have sort of achieved a more or less constant level of energy emission. A bigger earthquake, with ten times the power only happens 1/10th as often.. so over time if my thinking is right on this, the seismic region would be giving off essentially the same amount of energy per unit of time... does that make sense?

Oh well, I'll think on it while I ponder how they fit a logistic model to predict the mean amount of damage based on measurements of "roof drift".

Ok, just picked this off a site with statistics on the New Madrid fault (top photo), the most active fault in the US east of the Rockies, "Based upon historically and instrumentally recorded earthquakes, some scientists suggest the probability for a magnitude 6.0 or greater earthquake is 25-40% in the next 50 years and a 7-10% probability for a magnitude 7.5-8.0 within the next 50 years ".. Not quite the ten percent reduction for a increase of one level in magnitude. Hmmmm.. so that means there must be a very different b value for the New Madrid fault than they use for Southern California... does that make sense? Am I close in assuming that if the numbers above are correct, that we would make the b value for New Madrid to be about .6??? Help, not sure I'm putting all the pieces together right here.

Where Did the Analemma Go?

(Thanks to Lia C. for the pic)


The word is drawn from the Greek for a "lofty structure" or "upraised portion" of something, (a reference to the gnomen of a sundial) but the more common use of the term is to describe that figure eight that is, as I recently told me students, "on every globe."... Ooops... that seems not to be so.

It came from a lesson in trig on simple harmonic motion. We were talking about things that demonstrated sinusoidal behavior, and one bright young man suggested that the height of the sun at noon would be an example. I sort of agreed with a comment about "not exactly at noon.. but" and then the little guy was confused.. "You know, I said, like today ."(it was Feb 12) "I think the sun was about 12 minutes late or so."
Slow looks at each other, then back to me... the three letter word look,,,,,"Huh?"
"You know, that's what the analemma is for, telling if the sun is early or late.".....

Same look, compounded by the wild eye..."HUH?"

"On the Globe." Some frustration now... ".. the little figure eight... in the Pacific Ocean usually.. " Now the look has turned to fear... Will this be on the test? What is going on in the pacific? We were up doing homework, we don't watch the news... Help...and then, God answered their prayers...the bell rings... and they RUSH to exit...

So this morning, I came in and put out a call to the staff, "...someone loan me a globe, my kids don't know what an analemma is ". The first offer was just down the hall, cool, I'll walk down and get one and bring it in and when they come in I'll show them what it is and they'll say,... "Ohhhh, That thing." But it wasn't there... and it wasn't on the second globe offered, or the third... or ...gasp.. any of them.. the analemma has, it seems, gone the way of the two-dollar bill... So today I went in and set aside the basics of simple harmonic motion and we talked about a little astronomy, and why we have a 24 hour day.

First I pulled the graph of an Analemma off a Plus-Math on-line article from Cambridge, and checked some notes... (I was wrong about the globe..don't want to trust my aging memory)...
"The difference between the mean solar time and the actual solar time is called the equation of time. The Sun is furthest behind GMT around 12th February, when it is about 14 minutes 20 seconds slow. It is furthest ahead around 3rd November when it is about 16 minutes 23 seconds fast. The Sun's position coincides with GMT on four days of the year: 16th April, 14th June, 2nd September and 25th December," Later I found a cool picture taken by Dennis DiCicco, and editor with sky and telescope who spent two years taking pictures of the sun at noon once a week for a year (the second year was after he found out that the first had been slightly off center and cut off part of the bottom and had to start over...patience).

The Plus sight even a nice reminder that the "mean" in Greenwich Mean Time is really an average. This was a little more difficult for them to explain.

When I showed it to them not one reported ever seeing one. I asked them how long it took for the earth to spin on its axis and they mostly knew the memorized value...23 hrs 56 minutes (they always leave off the 4 seconds)... but they never thought about it.

Hmm, I ask.... "so if the sun reaches its highest point today at noon, and we turn around in 23 hours 56 min.. then tomorrow the high sun will be at 11:56..YES?" They conceded the four minutes possible error.

"And in two more weeks, the high sun would be about 11 am?????.. the hour is troublesome, but the math seems incontrovertible..so they mumble assent.
"And after a month,.... at ten am.... and in three months... The sun will reach its highest point at Six AM?????... Have you noticed that happening???" They were a little confused, but finally were ready to admit that the 23 hours 56 minute day must be a mistake..... Until one kid finds a reference.. "MR. Ballew, it says right here.. (finger poking into the book...we got that old guy now....."It says RIGHT here that the ..." and here we get a really bad attempt at "Sideral day or rotation period."

So we walk slowly through the idea that as the earth spins the very distant starts rotate in the sky and come back to their rightful position in 23 hours, 56 minutes and four seconds... but in that day, the earth also wandered part way around its elliptical path around the sun (god I will so love using this again when they are working with conics.. SURELY they will remember). And so the sun (a not so distant star) will have moved, and the earth will have to turn a little longer...about four minutes longer and till it is back to its rightful place high in the sky.... cool... but the figure eight thingy????

Well, remember Kepler??? "Equal areas in equal periods of time." But when we are farther from the sun, we move more slowly.. and so we turn too far.. the high sun comes early.... and when the sun is closer we move faster... and the sun shows up a little late... But the MEAN in Greenwich Mean Time...(we are big on GMT around here, not because we are in England, but this is a military base... Military Zulu time IS GMT) is really, the average of the time the sun will be at its highest. It is when noon comes around..."on average." Some times it is early, some times it is late, but on average a day from highest sun position to the next will be 24 hours.

We did manage to review a little for the trig stuff coming up, but they actually seemed to be thinking more than normal, offering conjectures and (too often) wild guesses. But mostly they were thinking, and doing a little math... and we all had a pretty good day...

Now.. can we Please get the analemma back on the globe?

Pure Math and Unintended Applications

Dave Bock sent a note to the AP Math EDG about this BBC audio lecture, and I wanted to share with a little larger audience...
Dave offered, "
Were a kid to ask "When does anybody ever use this stuff? ", here are some answers.
- Dave"

Don't ask how Dave hears about BBC programs before I do...the guy is just freaky that way...

In particular, in the first five minutes, Colva Roney-Dougal (who is a lecturer in Pure Mathematics at the University of St Andrews) explains how complex numbers influenced the birth of the electric chair (for all the wrong reasons) and became the reason we have A.C. current in the world and not D.C., as Edison hoped.

Then John Barrow, who is Professor of Mathematical Sciences at the University of Cambridge and the Gresham Lecturer in Geometry, will give a brief history of the conics that will provide detail for any Pre-calc course. Later Marcus DeSautoy will tell you about some Babylonian history that fits in any introduction to solving quadratic equations.

And all that is in the first twelve minutes... you still have 33 minutes of mystery left, and if you teach stats, Colva comes back to talk about De Moivre creating the Normal curve..

In the promotion on the BBC site they remind us that, "In his book The Mathematician's Apology (1941), the Cambridge mathematician G H Hardy expressed his reverence for pure maths, and celebrated its uselessness in the real world. Yet one of the branches of pure mathematics in which Hardy excelled was number theory, and it was this field which played a major role in the work of his younger colleague, Alan Turing, as he worked first to crack Nazi codes at Bletchley Park and then on one of the first computers."
Listen, share...

Meet Math in St. Louis

Happy Birthday, St. Louis (the city, not the guy)... It was on this day, Feb 15, 1764 that the city of St. Louis was founded.... I know, Wikipedia said so. Two hundred one years later, the Gateway Arch opened to the public in the Jefferson National Expansion Memorial. But for the math person, it is a big catenary arch... note to students, it is NOT a parabola... (OK, as you will see below, it is not an exact catenary)

Here is a video I found out about from Dave Richeson's Division by Zero blog that he posted a little while back about the math involved.


A little history for those, like me, who enjoy that sort of thing. It may be so perfectly fitting that the Jefferson National Expansion Memorial should have a catenary arch, since President Jefferson may have invented that exact English term for the shape. The shape is also sometimes called a chainette and a funicular curve from the Latin "funiculus" for a cord or rope.

For students of American History, it may be interesting that the first use of "catenary", rather than the longer, more formal "catenaria", may have been in a letter from Thomas Jefferson to Thomas Paine. Jeff Miller's wonderful web-site on the first use of mathematical words has

In a letter to Thomas Jefferson dated Sept. 15, 1788, Thomas Paine, discussing the design of a bridge, used the term catenarian arch:

Whether I shall set off a catenarian Arch or an Arch of a Circle I have not yet determined, but I mean to set off both and take my choice. There is one objection against a Catenarian Arch, which is, that the Iron tubes being all cast in one form will not exactly fit every part of it. An Arch of a Circle may be sett off to any extent by calculating the Ordinates, at equal distances on the diameter. In this case, the Radius will always be the Hypothenuse,(I believe that was the spelling in the letter, not my typo) the portion of the diameter be the Base, and the Ordinate the perpendicular or the Ordinate may be found by Trigonometry in which the Base, the Hypothenuse and right angle will be always given.

In a reply to Paine dated Dec. 23, 1788, Thomas Jefferson used the word catenary:

You hesitate between the catenary, and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches by the Abbe Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are that 'every part of the Catenary is in perfect equilibrium.'

The earliest citation for catenary in the OED2 is from the above letter.
============================================
Oh, and is anyone else surprised that Thomas Paine is involved in Bridge Design? I found another very brief comment from the same period that says: "1788: Mr. Paine resided at Rotherham in Yorkshire during part of the year, where an iron bridge upon the principle he had invented and laid before the Academy of Science was cast and erected."

In another article about the Walker Brothers Ironworks in Rotheram I found, "The Walkers were also known for their work with Thomas Paine from 1788 to 1791 in the construction of prototypical iron bridges. Thomas Paine, in addition to writing such works as The Rights of Man and Common Sense, was also one of the very earliest designers of single span iron bridges. In 1785 Paine tried unsuccessfully to gain support to erect an iron bridge over the Schuylkill River in Philadelphia. Unable to find investors in America, at Benjamin Franklin's suggestion Paine built a model of his design and traveled to England to show it to the Royal Society. Though his model was well received, still no backers were forthcoming, prompting Paine to seek out an ironworks which would built him a full scale prototype.
The Walker Ironworks in Rotherham agreed to do just that in the winter of 1788-1789. The bridge, with a span of 110 feet, was completed in 1790, but instead of being erected across the Thames River, as Paine would have liked, was instead set up in a field called Lisson Green, where other spectacles and attractions were located. Curious passers by could walk over the bridge or simply marvel at its construction. This endeavor, intended to attract possible investors in the building of pre-fabricated iron bridges, failed miserably. The rusting Lisson Green bridge was taken back to Rotherham by the Walkers in October of 1791. Thomas Paine, though disappointed, traveled to France where his writings about the French Revolution would land him in prison and nearly cost him his life.) The Walkers later built some of the earliest iron bridges ever made, including the bridge over the River Wear in Sunderland."

Now I wonder, was it (is it) circular or catenary?
And now I know...."The arch had a catenary shape and weighed 36½ tons, of which 23 tons was wrought iron and the remainder cast. It had a span of 110 feet and a rise of 5 feet. It remained on display for a year, attracting some fee-paying visitors but no offers to finance a Thames crossing. Eventually the abutments gave way—always a danger with a shallow arch—whereupon the bridge was dismantled and sent back to Rotherham. By then Paine had once again flung himself into revolutionary activities, this time in France, and his career as a bridge designer and promoter began a decade-long hiatus." From American Heritage Magazine

I should point out that this was not a singular stroll into science and engineering for the President. He also designed an improved moldboard (from Old English molde "earth, sand, dust, soil; land, country, world,") plow for which the French Society of Agriculture awarded Jefferson its gold medal and membership as a foreign associate.

Jeannie, Be My Valentine,

A little more public that I usually get with this kind of stuff...but some time you gotta go a little crazy... I love you Jeannie... say you'll be my valentine..... today, and forever..

Jason Aldean   More CMT Music   More CMT Music Videos

Casting about for the History of Casting Out Nines

Reviewing my blogging history lately, and was reminded of an unsuccessful search I had been on (and off) for two years. After all that time, I know exactly what I have below. If you have more to share, please do.
---------------------------
In my youth, back in the dark ages BC ( before calculators), one of the common mathematical tools we were taught was called casting out nines. It seems it is not common anymore, perhaps the errors students make with calculators no longer submit to the simple check of this ancient method. If you happen to be of the generation who have been kissed on the forehead by the Gods of Electronics, and don't know about this topic, you can find more on my notes, and also there is a brief video below although the use of "=" signs in some of these operations makes me go a little crazy.


Casting Out Nines Addition And Multiplication Math .

I have a note on my MathWords page on the subject from a respected math historian (Albrecht Heefer) that tells me, "Casting out nines is believed to be of Indian origin, but it does not occur before 950. Maximus Planudes called it 'Arithmetic after the Indian method". Along the way I seem to have a note from him, or someone in the same exchange of e-notes, telling me that I can find more confirmation on the web site of David Singmaster, the famous historian of mathematical recreation;.... but while searching there, I found a note that claims the first mention of casting out nines was by the Latin writer Iamblichus in 325 BC... But he(Iamblichus) was talking about Nichomachus, a Pythagorean who lived around 100 AD.Here is a quote from Dr. Singmasters page: "CASTING OUT NINES. This is often attributed to the Hindus, but I have some references to Greek special uses of it by St. Hippolytus (c200) and
Iamblichus (c325) though I haven't seen either source. I have read that Avicenna's attribution of it to the Hindus is a dubious interpretation. It
appears in al-Khw_rizm_ (c820) [this is a full century before the 950 date given earlier] and al-Uql_dis_ (952/953) as well as in Aryabhata II's Mah_-siddh_nta"


Now the common thought, or at least as I thought I understood it, was that the inventors of the Hindu-Arabic numerals had developed casting out nines and it sort of made its way into the west with the introduction of the Arabic numbers. Leonardo of Pisa, the famous Fibonacci whose bunny sequence you remember from school (of course you do, 1, 1, 2, 3, 5, 8, 13, 21...... That sequence) was a major influence in bringing both to the west with his famous book, the Liber Abaci, (the book of calculating) around 1202.


But the fact is that the general public held on to their Roman numerals for several centuries, and legal documents had to have them in some areas up into the 15th century.. Now the problem, at least for me, is that it seems much less likely that someone would develop casting out nines using Roman numerals.. see if you are using Arabic numerals, you take a number and add up the digits... 2534 would give 2+5+3+4 = 14 and then adding 1+4 = 5 so we know that if you divide 2534 by nine, you get a remainder of 5. Now in Roman numerals we write 2534 as MMDXXXIV ... It just looks less likely to jump out.. Ok, maybe if they never used the D for 500 and V for five, I can see it becoming obvious, so maybe that is how it came about. If you write the Roman numbers with only unit (that's how math types say ONE) multiplers, like M for 1000 or X for 10 or C for 100, then all you would have to do is count the number of digits (not add them up). For example MMCCXII has seven digits, so the number 2212 should have a digit root of 7, which it does. And for really long numbers, you could throw away groups of nine in the same way we do with casting out nines.....The 2534 we used before would appear as MMCCCCCXXXIIII, and we could cross out the MMCCCCCXX in front, count the XIIII at the end and say, ahhh, remainder of five.... MAYBE... but I wonder..

Anyway, I'm still looking for that Rogue Scholar out there who happens to have the original of Nicomachus' "Introduction to Arithmetic" laying around on his bookshelf and would like to translate for me to explain where he says it came from (if indeed he did).


So................... "Anyone? Anyone? ..... Bueller?" (too old for my students to get).

It Is Contagious..

Do you think we could catch it for out kids? I want this for the mission in my school, in my community...Aware, Enable, Empower...
Join me...

Nuts! Peanuts That is.

It was on a Sunday ten years ago today that the last original Peanuts hit the papers. Snoopy, atop his
doghouse at the typewriter, ended the run of a comic that had been there for a half-century. It happened, by chance, on the morning following the death of creator Charles Schulz.

I guess it is a day for nuts. Stockard Channing and Jerry Springer were both born on this day in 1944... and at least one of them is nuts. About 25 years before that, Tennesse Earnie Ford was born. He described himself as a pea-picker rather than a peanut, but close enough.

And in 1920, the National Negro League was founded by a nut from Chicago named Rube Foster. Rube was perhaps the greatest negro pitcher of the period...and he was the founder of the longest lasting of the negro leagues. When he started the league,he actually owned the contract of every player in the league. He managed to get the use of Mack Park in Detroit for one of his charter teams, the Detroit Stars. Mack Park was located about four miles from downtown Detroit at the corner of Fairview Ave. and Mack Ave, hence the name. Although the area was predominantly German immigrants, the Mack trolley made it easy for supporters to make it out to the park from the black areas of the city.

And then one day, as the story is told by Wikipedia, "In July 1929, the Kansas City Monarchs were in Detroit to play a doubleheader with the Stars. Two days of heavy rain left the ball field with standing water and threatened to postpone the game. Roesink,(the Grand Rapids hat maker who owned the park) working with the grounds crew, ordered gasoline to be spread on the field for eventual ignition to dry out the field and save the game from cancellation. (You know that was nuts..) After dispersing as much gasoline as they needed, the grounds crew stored the spare cans below the wooden bleachers(Uh oh... probably not a good idea). It is thought that a discarded cigarette butt accidentally ignited the gasoline on the field. Flames quickly spread to the storage area, resulting in a raging fire that engulfed the wooden framework of the stadium." (how totally unexpected..... if you are nuts.)

After the fire, Mack Park was rebuilt and managed to provide a site for HS baseball until in the sixties, then an influx of federal money turned it into a home for the elderly. There are some nice looking condo-like apartments there...but it isn't a very good place to play baseball.

And so the Stars moved to Hamtramack. By 1931 the depression had brought the Negro National League to an end. The Stars continued playing as an independent team, and then as part of various short lived leagues. The team was still alive in 1958 when the owner, Ted Rasberry, decided to rename the team after former negro league star Goose Tatum... Now there is a real nut, but you probably know the Goose better for his antics on the Basketball court. He was the original "Clown Prince" of the Harlem Globetrotters. Tatum started his career in the 1940s as a baseball player for the Birmingham Black Barons and the Indianapolis Clowns of the (new)Negro National League. It was during this time that he started clowning around on the field to amuse the crowds. Abe Saperstein spotted Tatum clowning around on a baseball field and put him to work. Not so nutty, he invented the "sky hook" that would make Kareem Jabbar one of the most potent offensive forces in both college and professional basketball.

The Stars lived only two more years, and the Goose lived seven more after that.

And this was the day on which the celebration of Lupercalia began in Ancient Rome. The Holiday derived from the festival of Februa, from which this month gets its name. It came originally from the Sabine tribes, an ancient tribe from central Italy who were conquered by the Romans around 290 BC. The festival was a fertility ritual in which the women were flogged with an appendage, or organ, of an animal (sorry, I'm not sure what,,,or I'm not bold enough to tell you what) with the supposed result that they would be more likely to bear children. It is through association with this festival that the romantic associations of St. Valentine's Day began, a day that originally had no association with love or relationships. And the theme today is....... yeah, that's nuts.

For the math history freaks, you know who you are.. today is also the birthday of Peter Gustav Dirichlet, the German mathemtician, who was born on Feb 13,1805. Dirichlet is remembered for a theorem he used in working on Pell's equation, which has more recently become known as the Pigeon-hole theorem. It seems that term was originate by Paul Erdos in a 1956 paper. Before that... well here is a note I have from a not-too recent discussion on a history group by Julio Cabillon. He added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portugese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish we use also:"the principle of the drawers of Dirichlet" that is 'Zasada szufladkowa Dirichleta' ". Dirichlet first wrote about it in " Recherches sur les formes quadratiques à coefficients et à indéterminées complexes" (J. reine u. angew. Math. (24 (1842) 291 371) = Math. Werke, (1889 1897), which was reprinted by Chelsea, 1969, vol. I, pp. 533 618. On pp. 579 580, he uses the principle to find good rational approximations. He doesn't give it a name. In later works he called it the "Schubfach Prinzip" [which I am told means "drawer principle" in German]

I had assumed, as stated on the Wolfram "MathWorld" site, that,"This statement has important applications in number theory and was first stated by Dirichlet in 1834". In truth, the principal has been around much longer than Dirichlet, as I found out in June of 2009 when Dave Renfro sent me word that the idea pops up in the unexpected (at least by me) work, "Portraits of the seventeenth century, historic and literary", by Charles Augustin Sainte-Beuve. During his description of Mme. de Longuevillle, who was Ann-Genevieve De Bourbon, and lived from 1619 to 1679 he tells the following story:
"I asked M. Nicole (See below for description of M. Nicole) one day what was the character of Mme. de Longueville's mind; he told me she had a very keen and very delicate mind in knowledge of the character of individuals, but that it was very small, very weak, very limited on matters of science and reasoning, and on all speculative matters in which there was no question of sentiment ' For example,' added he, ' I told her one day that I could bet and prove that there were in Paris at least two inhabitants who had the same number of hairs upon their head, though I could not point out who were those two persons. She said i could not be certain of it until I had counted the hairs of the two persons. Here is my demonstration/ I said to her: M lay it down as a fact that the best-fiimbhed head does not possess more than 200,000 hairs, and the most scantily furnished head b that which has only 1 hair. If, now you suppose that 200,000 heads all have a different number of hairs, they must each have one of the numbers of hairs which are between 1 and 200,000; for if we suppose that there were 2 among these 200,000 who had the same number of hairs, I win my bet But suppose these 200,000 inhabitants all have a different number of hairs, if I bring in a single other inhabitant who has hairs and has no more than 200,000 of them, it necessarily follows that this number of hairs, whatever it be, will be found between 1 and 200,000, and, consequently, be equal in number of hairs to one of the 200,000 heads. Now, as instead of one inhabitant more than 200,000, there are, in all, nearly 800,000 inhabitants in Paris, you see plainly that there must be many heads equal in number of hairs, although I have not counted them.' Mme. de Longuevillle still could not understand that demonstration could be made of the equality in number of hairs, and she always maintained that the only way to prove it was to count them. "
The M. Nicole who demonstrated the principal was Pierre Nicole, (1625 -1695), one of the most distinguished of the French Jansenist writers, sometimes compared more favorably than Pascal for his writings on the moral reasoning of the Port Royal Jansenist. It may be that he had picked up the principal from Antoine Arnauld, another Port Royal Jansenist who was an influential mathematician and logician.

It is the kind of tool you need if you want to prove that there are two people living in Detroit who have exactly the same number of hairs on their head. (honest...no really, that is NOT nuts).

Fort Worth Snowman

I grew up in Fort Worth, and if you took all the snow that fell in my yard for all the winters that I lived there.... there would not be enough to build this snowman...reported as 10 to 12 foot high (that's as high or higher than the goal on a basketball backboard).... Somebody is Messing with Texas... Glad to be an ex-Texan...

The meaning of February 11..

Feb 11th is the 42 day of the year, and everyone knows that 42 is the answer to the question, "What is the meaning of life, the universe, and everything?"; at least according to Douglas Adams, Hitchhikers Guide to the Universe. And if you don't believe me, just ask WolframAlpha....

42 is interesting on its own without Mr. Adams. The binary expression is cute, 101010; and it is one of those Pythagorean 3D type numbers... its the sum of three squares, 1² + 4²+5². It is also the fifth Catalan Number.

Today is the day on which Bernard le Bovier de Fontenelle, French Scientist and writer, was born in in 1657.. He is worthy of remembering for one classic line...When, in his late nineties he met the beautiful Mme Helvétius, he reportedly told her, "Ah Madame, if only I were eighty again!". He was born exactly seven years after Rene Descartes died.

William Henry Fox Talbot, who invented the photographic negative, was born on this day in 1800... and you may not remember him, but his home in the Cotswold's was the scene for Harry Potter's school in the movies.

This is also the birth date of Josiah Willard Gibbs in 1839. Gibbs was a mathematician/chemist/physicist who laid the foundation for thermodynamics and invented vector analysis. Yale made him the first American to earn a PhD in Engineering. . He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis.

Today is also Inventor's Day in the USA.... but I am not making all this up.

Archimedes and Calculus....

I showed my calculus kids that Archimedes knew the area between the curve y=x² and the x-axis from x=-2 to x=2, which is one of those really early things you evaluate as you learn the wonders of the definite integral. Actually, what he knew was the area between the curve y=4-x² and the x-axis; and he knew how to subtract one area from another. I pointed out to my students that it was an easy extension of the triangle area formula A=1/2 bh. To find the area inside a parabola which was cut by a chord perpendicular to the axis of symmetry. You just have to change the constant... and the area is 2/3 b h where the base is the length of the chord and the height is the perpendicular distance from the chord to the vertex of the parabola.


Now Archimedes actually knew more (lots more) than that... for example he knew that the same rule applied to any chord through a parabola if you measured the height as the greatest perpendicular distance from the chord to the arc of the parabola.

I’ve been researching a little about his writing, and can add that to figure out the above, he became the first person to evaluate an infinite geometric series(or at least the first to do it in writing and leave it where we could find it).

What Archimedes actually learned was that if you drew a chord to a parabola, the area of the section between the chord and the parabola is 4/3 the area of the largest inscribed triangle with the chord as a base.( and the triangle is 1/2 b h so 1/2 of 4/3 gives the 2/3... focus children).. In the picture I have shown the graph of y = 4-x² cut by the chord y=x+2 as an example.


The triangle has vertices at (-2,0), (1,3), and (-.5, 3.75){side bar problem.... can you show that the greatest perpendicular distance between the chord and the parabola will always be at a point where the tangent to the parabola was the same as the slope of the chord?}. He did this by showing that if you inscribed two more triangles in the sections of the parabola outside two sides of the triangle given, that they would add up to ¼ of the given triangle. And then if you draw four more outside these two, they will add up to 1/16 of the first. He then set out to show that 1 + ¼ + 1/16 + 1/64…. = 4/3. 1800 years before the invention of calculus, he used the idea of a limit to show

1 + 1/3 (1) = 4/3

1 + ¼ + 1/3 (1/4) = 4/3

1 + ¼ + 1/16 + 1/3 (1/16) = 4/3 and extended this to show

1 + ¼ + 1/16 + … 1/ (4n) + 1/3 (1/(4n) = 4/3 and of course, as n goes to infinity,1/(4n) also goes to zero, leaving the sum 4/3.


Simply incredible.... Archimedes....wow, .. Just know children.....He was NOT like us.

A Little Geometry Music Please

Just for a kick, I thought I would pose a math problem.

Suppose you pick n points equally spaced around a circle of Radius R. Now pick a number relatively prime to n, call it p, and starting at some point on the circle, draw the chord connecting that point to the point p steps away around the perimeter. Start at that point and repeat the process with the point p steps away again.. keep going and eventually you come back to the point where you started. In the image below, n=12 and p = 5. Now draw the new circle that is tangent to all the chords. Call the radius of this circle r. Now, what is the ratio of the two radii, r/R?





You can download a file here that will draw this for up to n=19 (of course you can adjust the slider to make more). You have to have geogebra downloaded first (its free). First solution wins a shiny "attaboy" (or atta-person for the gender sensitive).

When I took those old Woodrow Wilson summer programs these were called "star polygons". The one in the picture is called the {12/5} star polygon.. This notation is often called a Schläfli symbol after the 19th-century mathematician Ludwig Schläfli. For example, you can indicate an equilateral triangle by {3} or a square by {4}. The classic pentagram (the star you draw without picking up the pencil, is the {5/2} star polygon. Of course {5,3} would be the same figure. An interesting challenge for students is to find the angle at any vertex of {n/p} as a function of n and p.

Ummm........ NO!!!!

I Stole this from the Facebook page of Dave Bock (noted Statistics textbook author) and loved the comment by one of his friends, Karen Seifert, who wondered, "How much would it cost to get your response (Um... NO!!!) on the next Billboard?"

Dave says the billboard is on the I-35 in Minnesota... shame Minnesota, shame!

So who can top that one?? Send your suggestions.

Close order Drill and the square root of -1

Replay from a blog about two years ago, requested by a student who was mentioned indirectly.,..

I’ve been thinking about groups lately, both the mathematical kind and the social kind. The thoughts were prompted by another trip with some of my kids to the Center for Mathematical Sciences in Cambridge to hear Professor Marcus de Sautoy speak on his new book, Finding Moonshine, which is soon to be released [The American title seems to be a little different] and is about symmetry and mathematical group theory. When mathematicians talk about symmetry, we talk about groups.

The social group that joins me on my trips seems diverse at first glance. There are a few seniors, a few juniors, and a smattering of sophomores. There are about the same number of boys and girls, and they are as different as high school boys and girls can be. The girls will giggle at themselves, amazed that they can be excited about going to a math lecture, while the boys grow more quiet than usual when they are out of their natural turf. We stand outside one of the lecture rooms and study a board covered with symbols that neither they nor I can decipher. They try to recognize some of the symbols, a “!” factorial symbol here, “and” and “or” logic symbols there. “Is that Sigma for summation?” Some one suggests its some kind of probability problem. “What’s that one, With the Pi in the parenthesis?” Grateful that I actually know one they don’t, I explain, “That’s a capital Gamma, a function like factorial that works for all real values, not just integers.” They say “Ahh” as if they understand. We’ll expand on that on another trip perhaps, a day when an expert can peal back another level of the mathematical mystery just a bit.

What makes some kids, some people, open their minds to the complexities of math, drawn to the cryptic symbols they don’t understand? It must be the same drive that led Champollion to decipher the Hieroglyphs. How is it that one kid can be blinded to the relationship of imaginary numbers by the simple hurdle of its name, while another wants to visualize a snowflake that exists in a universe with 196,883 dimensions.

All these thoughts wandered through my mind as I watched a group of ROTC students passing beneath my window and stop at attention in the open courtyard below. As they practices the simple stationary turns common to such formations, “A’ Ten Hut”, “Right Face”, About Face”, “Left Face”, I realized that I was watching them perform a physical demonstration of the same relations they would swear were too complex to understand in their algebra classes.

The mathematical term Isomorphism is from the Greek roots for “same body”. The parade ground moves represented an order four group that was isomorphic to the multiplicative relationships between the imaginary quantities they found so impossible to comprehend. The four activities on the drill pad could each be paired with the four primitive mathematical quantities, 1, -1, i, and –i, so that they each would produce the same result.

One is the identity, Like “Attention” it keeps the position fixed. About face is the same as -1. If we think of right face as i, the square root of -1, then left face would be –i, the opposite of i. Any two actions on the pad operated like multiplication of its counterparts in the abstract number set. About face followed by about face was like Attention, not turning at all, just as multiplying -1 by -1 returned us to the mathematical identity. Right face followed by about face produced left face, and mathematically -1 x I gives –i. What about that mysterious i x i that so confused them in the algebra class. Right face followed by right face was just about face, the drill pad’s symbol for -1. No student in the small squad in front of me would have thought it difficult to imagine what his position would be if I asked him how he would be standing if he made back to back right faces.

The same abstractions that made math so powerful, that allowed us to represent the drill team, and the multiplication of complex numbers with the same symbols was a beauty they could not see. But some will twist their faces and squint against the dimensional curse of being born in three-space, hoping to get a glimpse of a four-space hypercube, one step closer to the Monster. Only 196,879 more dimensions to climb. But on the way home, they start with baby steps. Someone asks, “Can there be a group with just two elements?’ I was going to answer, when one conjectures, “Yeah, I think just one and negative one would be a group under multiplication.” They discuss it for awhile, ignoring the old man driving the car; after all, they learned tonight that “a mathematician seldom produces great work over the age of forty”, and if they don’t know exactly how old I am, they know that my age is much greater than forty; or as they would say, my age =

Yep, perhaps too old to produce great math... but now and then I still turn out a great student... even after forty.. and that will be enough to keep me going for awhile.. thanks kids...

More Good Science

A simple question, "What can you hold in your hands?".... and wow, some answer.

"I have seen firsthand that mothers will do anything they can to save their babies. And yet 450 babies die every hour around the world. In villages where this toll is a reality, it is simply beyond a mother’s means to save her children."
Jane Chen



Now do something...wipe the tear from your eye, pull out your credit card, click on this link, and change your world... THAT's what you hold in your hands.

e-day and Andy Jackson

Adding on to the post about coincidences yesterday, this post is about a value derived from the same hyperbola, y=1/x. For the mathematician, February 7th, (or 2 - 7) is the date we decide to celebrate the constant which is the base of the natural logarithms, appx 2.71828.... (more later, and a way to memorize it). There have been LOTS of sites that explain LOTS of things (such as at Homeschool Math Blog and here at Let's Play Math.) about the value e, so I will try to not be too redundant and throw in something totally different, as the Monty Python folks used to say...


The letter e was first used for the base of the natural or hyperbolic logarithms by Leonhard Euler. Earlier I had mistakenly thought that Euler was the discoverer of the value, but in fact the number was published in Edward Wright's English translation of Napier's work on logarithms in 1618, almost 100 years before Euler's birth. [and in fact, it was known to the English Mathematician Roger Cotes. Cotes is one of those many promising mathematicians who died at a young age and Newton, who seldom said anything good about anyone else, once said "Perhaps if Cotes had lived, we would have known something"..] The number represented by e is approximately 2.718281828459045... Euler actually computed the number to eight more decimal places. This was done in 1727, and would seem almost impossible accuracy for anyone else, but of Euler it was said, "Euler calculates as other men breathe."

It was known from the work of Gregory of St. Vincent and others that the logarithms were somehow linked to the area under the hyperbola f(x)=1/x because the area under the curve matched the logarithmic property Log(AB)= Log(A)+Log(B). The Area under the curve from 1 to x=ab is equal to the areas from 1 to x=a plus the area from 1 to x=b. The value of e is such that the area under the hyperbola from 1 to e is 1 square unit. It has been conjectured that Euler may have used e as an abbreviation of the word Eins, the German word for one.

One oddity that students and teachers may use to remember the first 15 digits of e, given above, is to recognize their relationship with Andrew Jackson's presidency and an isosceles right triangle. Confusing? Just wait, all will be clear. We begin with 2, because Jackson was president for two terms. The 7 tells us he was the seventh president of the US. 1828 is the year he was elected, and we repeat this because of the two terms. Then we give the three angles of an isosceles right triangle, 45, 90, 45, and we have completed 15 digits of the base of the natural logarithms. I am almost 100% sure I picked that up from one of Martin Gardner's Scientific American columns.

Euler was one of the most influential mathematicians of the period and his prestige was sufficient that his use of a variable often marked it for posterity, but there were other symbols that were suggested occasionally. D'Alembert used c for the same constant in 1747, and Benjamin Peirce suggested a symbol that looked like a paper clip, or the @ symbol now used for e-mail addresses instead of pi, and the same symbol reflected in a vertical line for e.


But now I have to bring up the fact that in my new home town(My lovely Jeannie is there right now working on our "Boat House" which will be our retirement home out of the snow of Michigan.
)of Paducah, Ky, e-day is for Engineers Day. The University of Kentucky College of Engineering has a branch campus at Paducah,and they are having their open house on February 21 at Crounse Hall. They have, among other things, an Edible car contest (would I kid you?) as well as an Egg Drop contest, A Popsicle Stick bridge contest, and of course (drum roll please...) A Duct Tape Challenge...... I had a student only a few years ago who was a master of duct-tape-utilization. He would make roses out of duct tape to impress the ladies, (and did) and had a duct tape wallet... and once came to school in a sport coat made entirely of duct tape... I imagine he could have had one in cashmere for about the same price.

Concurrencies and Coincidences

Steve Phelps, over at concurrencies, just wrote a "What can you do with three random points in the plane?" blog. Coincidentally, I had just finished an interesting old (1902) article about three random points on an equilateral hyperbola (such as y= 1/x for those not familiar with the term). And by another coincidence, the article happened to involve one of the common concurrent centers of a triangle, the orthocenter where the three altitudes from the vertices intersects.

It turns out, that if you pick three random points on a equilateral hyperbola (they can be on either branch), then the orthocenter will also fall on the hyperbola. Stated another way, if you pick the three points all on one branch and make them all free to move, the locus of the orthocenter will be the other branch of the hyperbola.

Poncelet had actually written about this as far back as Jan of 1821 in Gergonne's Annales.

Oh, by the way, a little "prove this factoid" for my calc kids... the y-intercept of the tangent line to any point on the rectangular hyperbola is always twice the y-coordinate, and the slope is always the square of the reciprocal of the y-coordinate. SWEET! [Yikes, I've been busted... Keninwa noticed a mistake in the above (thanks guy) actually what I should have said (and this is only true for the basic y=1/x case), the slope is equal to the negative of the square of the y=value .. (and now, head hanging in shame, he wanders off into the sunset, muttering to himself about proofreading)..

Triangle Area Formulas, Old and New

Ok, I know Heron's Formula (if your teacher calls it Hero's formula, it's the same)...

Heron of Alexandria, sometimes called Hero, lived around the year 100 AD and is most often remembered for a formula for the area of a triangle. The formula gives a method of computing the area from the lengths of the three sides. If we call the sides a, b, and c; then the area is given by where the "s" stands for the semi-perimeter, . You can find a nice geometric proof of Heron's formula at this link to the Dr. Math site. The proof was done by Dr. Floor, who credits the method to Paul Yiu of Florida Atlantic University.


Documents from the Arabic writers indicate Archimedes may well have known this formula 300 years before Heron. In 1896 a copy of Heron's Metrica was recovered in Constantinople (now Istanbul) that had been copied around 1100 AD. It contains the oldest known demonstration of the formula. Heron is also remembered for his invention of a primitive steam engine and one of the earliest forerunners of the thermometer. The image at right shows a picture of a reconstruction of Heron's steam engine. The image is from the Smith College meuseum of Ancient Inventions where you can find more about Heron's, and many other's, interesting creations.
An extension of Heron's area formula for cyclic quadrilaterals is known as Brahmagupta's Formula

Heron's Metrica also contains one of the earliest examples of a method of finding square roots that is called the divide and average model. To find an approximate square root of a number, N, think of any number smaller than N, which we will call M. Then find a new approximation by letting E = (M + N/M)/2. Another approximation can be found by repeating the method with this new approximation. For example, beginning with N=20 and M= 2, we get E= (2 + 20/2) / 2 or E= (2+10)/2 = 6.
Repeating with M= 6 we get E= (6+ 20/6)/2 = ( 6 + 3 1/3 )/2 = 14/3 or 4 2/3. After only two iterations from a very bad starting guess the approximation is within .2 of the correct value.

Heron is also remembered for a problem he solved in Catoprica; Given two points, A and B, on the same side of a line, find a point X on the line so that the total distance AX+XB is a minimum. The solution may come quickly if you know that the translation of Catoprica is "About Mirrors". The solution given by Heron is to find the mirror reflection of point B in the line, B', and draw a straight line from A to B'. Where it intersects the line is the choice of point X.


Ok, so much for the old news... but recently I was going through some old journals that Dave Refro sends me from time to time to keep me out of mischief, and I came across an article in the 1885 Annals of Mathematics which listed 105 different formulas for the area of a triangle ( things to do on a rainy afternoon, list 110 different formulae for the area of a triangle). One was the well known Heron's formula above and then there was another that looked strikingly similar. If we let M_A be the length of the median to vertex A, and similarly for M_B and M_C . The we can call sigma 1/2 the sum of M_A + M_Ba+M_C. Then we can write.

Now that is a new one to me..