Solving Geometric Problem with Pure Logic

I recently came across one of those geometry puzzles with multiple choice answers people like to post.

It struck me that a clever student could figure out the answer without calculations.  Before I go on I'll post the puzzle and let you play at it, but no calculating.  

You see these things all over the place if you are on the internet much.  This one at least was more straightforward than the "95% of people can't do this arithmetic problem" and the do something with order of operations using an obelus  (the spit symbol, ÷, used only in English speaking countries for division, strangely first used in a German book, Teutsche Algebra by Johann Rahn.)  

Anyway, that diversion was to take up a little space between the problem and the answer.

Since I knew the answer I read some of the different approaches in the comments. My favorite was a guy whose solution was pure logic.  

Answers a) and c) rule each other out,  since sides b and  a are the legs of the right triangle, if you switched their names, you get a different radius for congruent right triangles.  Answer b)  can't be right either.  The incircle can not be longer than the hypotenuse, because it is inside the triangle, but in any triangle, the two shorter sides are more than the longest side, which means that the b) answer woud be greater than the hypotenuse.  

So if any of them are true, d) must be the one.  

And it is true, but I'm too lazy to write it out, here is a  solution I plucked from  Quora.  

That's the kind of clever thinking that makes teachers smile.

The Cubic Attractiveness of 153

From 2012 post, with additional detail and a guest followup:

Just in time for June 1st, which is the 153rd day of this leap year.  I recently discovered an interesting quality about the number 153.

Ok, I was amazed. I was lead from a comment (below) that this is all shown in the online integer sequence, but it is still great fun.

Pick any old number you want, and multiply by three (or just pick a number that is a multiple of three).
Now take all the digits and cube them and add the cubes together.
For example, if you picked 231, you would add 2³ + 3 ³ + 1 ³ to get 36. 
Yeah, So what you're probably thinking... but take that new number and do the same thing... cube the digits and add them up... Nothing?  Keep going... eventually you get to 153, and then when you  do it again, you get 153 forever.
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A digression about notation and terminology
In slightly more formal language, it seems that 153 is the fixed point attractor of any multiple of three under the process of summing the cubes of the digits. This process of an n-digit number, k,  which has a sun of the nth power of its digits equal to k, is variously called a narcissistic number, an Armstrong number, or a perfect digital invariant(often written PDI and PPDI).  So 153 is a Narcissistic number.  The other three digit numbers that are narcisscistic are 370, 371, and 407.  

In his famous A Mathematicians Apology, G, H. Hardy dismisses these numbers with the comment that, "There is nothing in these odd facts which appeals to the mathematician."  

But recreational mathematicians move on undisturbed by Hardy's judgement.  The first use of Narcissistic numbers that I have found, although defined a little more broadly than is now done, was in  Joseph S. Madachy's Mathematics on Vacation in 1966. He described them as a number that is equal to a function of its digits, and included numbers like 145 = 1! + 4! + 5! 

The earliest record of Armstrong Numbers was in a 1971 book,  Computer Science Laboratory Exercises  by F. D. Federighi, ‎Edwin D. Reilly . I suspect much of the popularity of the numbers sprang from being a good programming exercise.

END Digression

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The 231 that I used above goes to 36, which goes to 243, then 99, 1458, 702 and 351, which give 153. 

If you started with something like 72, it gets there pretty quickly,  72 →   351 →  153.  

Other numbers take a little longer.  717, for example, takes 13 iterations of the cubing process to get there...but it does.  
And 153 is special... ONLY the multiples of three go there.  Pick another number that's not a multiple of three and run the process and several things might happen, but what won't happen, is going to 153. 

When I began to explore why this happened I could figure out a few things easy enough.  The first realization was that the iterations couldn't just get bigger and bigger and diverge to infinity.  9³ is 729 and so any four digit number has to iterate to a value less than 4 * 729 or 2916, and anything with more digits just keeps getting pushed down until it gets under that limit.
So all the numbers in the world have to end up doing something other than just keep getting bigger.
That only leaves a few options.  They might just keep jumping around in some orbit that cycles through several numbers.  This happens with 46 for example.  After a few iterations you fall into a three cycle. 

46 → 280 → 520 →  133 → 55  → 250 →133

Other numbers also do this, and there are a few different three cycles and  two cycles, but that is sort of an oddity, at least for numbers less than 1000 (still have work to do here).
Most numbers go to a fixed aatractor.  For multiples of three, that seems to always be 153.   For numbers that are not multiples of three, the general fixed points are either 370, or 371.  And they have a modulo three relationship as well.  Numbers that are congruent to 2 Mod₃ (they have a remainder of 2 when you divide by three) generally go to 371.  The exceptions are a couple of numbers; 47 , 74, 77,  89 & 98,   which go to 407 (of course 707, 908, 980, etc would also, I counted 30 numbers less than 1000 that go to 407 as a fixed point).  

All the cycles that I have found are numbers that are equivalent to 1 Mod₃.  The cycles seem pretty common with less than 1/2 the smaller numbers going to a fixed value of 370 and the occasional few like 1, 10, 100, that have a fixed point at one. Similar to the Happy numbers under the squared sum.  118 is in that group as well, for example.  

The reason for the separation into modulus classes of three is easy enough to explain.  When you cube a number, it's modulus in base three isn't changed.  For example, 4 Mod₃ is 1, and 4³ = 64 is also equivalent to 1 Mod₃.  So the Modulus of a number doesn't change under this process, grouping the results together.  

Now when you add in the fact that there are not really that many of these smaller (less than 3000, say) numbers that can be made with the sum of cubes unless you allow for weird numbers like 11,111 or something to get five.  

I haven't explored much beyond 1000, so I'm not sure if I will come across other cyclic orbits. Or why only the 3n+1 type numbers produce cycles. And I'm not sure what else I may find, but I'm thinking that what I've seen so far makes 153 a very special number. 

After I wrote this back in 2012, a young friend working on his degree at Pitt sent me notes on his reaults when he  took off on the fourth power,  He surmised that all the numbers two through nine went to a fixed value of 13139 and then repeated the cycle (13139 --> 6725 --> 4338 --> 4514 --> 1138 --> 4179 --> 9219 --> 13139...)  I checked this for a couple and it took a few iterations to get there, but the ones I checked did.  That means numbers like 11, 20, 30, .. 101, 111, 200,  and similar numbers if digits sum to 2-9.

He also found that 12 had a fixed point attractor at 8208, ( 12 --> 17 --> 2402 --> 288 --> 8208 --> 8208 --> 8208 ) which means that 17, 21, 71, 102, 107, 170, 201, 210, 224, 242, 288, 422, 701, 710,828, 882 will all go to the same attractor (and of course many more).  

I have now convinced myself that 13, 14, 15, 16, 18, 19,   go to the same cycle as 2 through 9 so now 2 through 9, 11, 13, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26,27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 50, 51, 52, 56, 62, 65, 72, 81, 91 101, 111, 200...

All end in the cycle (13139 --> 6725 --> 4338 --> 4514 --> 1138 --> 4179 --> 9219 --> 13139...

If my calculating does not contain any big errors, so far I have found end behavior for the following numbers under 50:

Two  have fixed attractors at 1 (1,10)

forty (listed above) end at the cycle (13139 --> 6725 --> 4338 --> 4514 --> 1138 --> 4179 --> 9219 --> 13139...

Three end in the fixed point 8208 (12, 17, 21 )  

5 of the numbers are not yet checked. In time I will reduce these. 

A little over a year after I did this, Derek had spent more time working on this and I asked him to guest post his work in progress. 

So here are his notes (I'll add notes of what I have found over the last decade in italics along the way):

Digits to the Fourth Power

By Derek J. Orr, University of Pittsburgh

I was perusing through Pat’s blog here and I noticed two pages that involved summing the squares of digits and summing the cubes of digits. I, then, proposed taking the fourth power of these digits. When I did so, I found some pretty interesting results. Unfortunately, it is time consuming to do each of these numbers manually but I plan to get up to 1,000 and maybe even 10,000 (if motivation and free time permit).

Some numbers that I found went to 1 and are called “happy numbers”. Unfortunately, the only ones I’ve found were 1, 10 and 100 (I went up to 225 so far). However, I am assuming the next smallest happy number is 1,000 just because it’s hard to find one. I wanted to find one and, since I couldn’t, I forced myself to.

I know that a happy number is a number that will eventually reduce to the number 1 and repeat forever. So, the guaranteed happy numbers are 1, 10, 100, 1000, 10000, etc. Thus, I figured out what numbers can get me to these guaranteed happy numbers and, though they aren’t small, I found a few. First, I’ll write out the fourth powers of the single digit numbers: 

Using these values, I found different combinations that could get me to the guaranteed happy numbers. Also, since we have 1 as a possibility, it’s impossible to skip any number I choose. The table below (above) lists a few combinations that I found.

The numbers inside the table represent how many of the digits we need. So, for the first line of the table, the number 1,111,111,111 is a happy number because added together 10 times gets us 10. From this table, the smallest happy number that is not a multiple of 10 (and that is not 1) is 11,123. However, in just two steps, this number can reach 1. What if there are longer iterations that still get us to 1? What if there is a number that can get us 11,123? So, I experimented:

(the other equation I found has 18 digits, so I won’t write it out)

So, we see that there is a new happy number, 22,233,489. Once again, what numbers get to this number? I could do this all day but I won’t, mainly because 22,233,489 can be divided by over 3,388 times. So, this means that the next number has more than 3,388 digits, which is far too many. I’ve assumed that up to now, the numbers will only get bigger. However, what if we tried a different number instead of 11,123? Since the digits are added, we can always change the order. So, 12113, 13112, 21113, 31112, 12131, 13121, 12311, 13211, etc. are also happy numbers. Again, I could experiment on these but I won’t just to save time. I believe it is safe to say that the smallest happy number without a zero (and that is not 1) is 11,123. Since the happy numbers are so hard to find, I looked at different numbers.

One loop that I found was with the number 2,178. I saw it for the first time when I tried the number 127. Here is the iteration for 127.

127 -- 2418 -- 4369 -- 8194 --10914 -- 6819 -- 11954 -- 7444 -- 3169 -- 7939 -- 15604 -- 2178 -- 6514 -- 2178 -- 6514 --…

So, we can see it goes through this 2,178----6,514 loop. Now, I’ve only seen this work for 127, 172, and 217 (so far) but the four- and five-digit numbers above will also bring about this loop.

Also, I found that there seems to be a fixed point where some numbers end up at, similar to what Pat found when cubing the digits. This number was 8,208, and satisfies the condition (ie 8^4 + 2^4 + 0^4 + 8^4 = 8208) . What numbers gave me 8,208? I computed a bit more than the first 200 digits (225 to be exact) and found the numbers 12, 17, 21, 46, 64, 71, 102, 107, 120, 137, 145, 154, 170, 173, 201, 210, and 224 (and of course 288) will get to 8,208 and stay there. Now, past these, it’s obvious that 317, 415, 514, 710, 701, 713, 422, etc. also work. When you cubed the digits, Pat found that you reach the number 153 and it repeats forever. However, Pat found that if you have any multiple of three, you will reach 153. There is a pattern there. I am sadly not able to find any pattern with these; they seem to be random, like the happy numbers when you square the digits (1, 7, 10, 13, 19, 23, 28…etc.).

Other four digit 

Another loop number these could go to is 13,139. With 13,139, there is a loop involved (13139 -- 6725 -- 4338 -- 4514 -- 1138 -- 4179 -- 9219 -- 13139…). This has happened with every number I haven’t mentioned (around 90% of the numbers I’ve tried).(lots of numbers take a very long time to drop into their cycle)  

Going back to 8208, I keep wondering if there is another fixed point. When cubing the digits, there are five numbers that equal the sum of their digits cubed: 1, 153, 370, 371, and 407. When squaring the digits, there is only one number that equals the sum of its digits squared: 1. But, when taking the fourth power, I have only found two numbers that work: 1 and 8,208.  I do wonder if there are more or not; perhaps a good problem for a computer programmer because doing these manually, though possible, takes up a lot of time. (Derek didn't find the other two fixed point attractors, 1634 and 9474)

It would seem that the only sum of four fourth powers that sum to 1634 are the actual digits of 1,6,3 and 4. My reasoning is that there can not be any digit greater than 6, since the fourth powers of 7,8, and 9 all exceed 1634.  No four digits picked from 0, 1, 2, 3, and 4 can have a sum large enough since 1634 exceeds 4 times 4^4 .  So there must be at least one five or one six, but not one of each , nor two sixes.  So we need only to check using one six and three less than 5, or one or two fives with the rest less than five. trying each of these combinations fails.

9474 seems to suffer a similar fate of attracting only the numbers made up of the same four digits.

 ---------------------------------------

I will mention one oddity I came across over the years playing with these, the palindrome 11^4 =14641  is the smallest fourth power whose digits sum to a fourth power, 16.

I haven't pursued the behavior of the fifth power of these, but have been told that the fixed point attractors are 1, 4150, 4151, 54748, 92727, 93084, 194979

If you are one of those number curious folks who also happens to be a really good programmer, I would love to receive a list of other fixed points and cycles in third, fourth, and fifth powers of digits. (Or maybe you can send your own guest blog.)

The Distracted Goalie

  

The great Physicist, Niels Bohr, was brother to an outstanding mathematician, Harald who founded the field of almost periodic functions. In their youth both were very good athletes, with Harald clearly the more dedicated sportsman. Harald had been a member of the Danish National Football team while still a student and had earned a silver medal as such in the 1908 Olympic games; the first time the Olympic games had football. Harald scored two goals in the opening game defeating the French nine-zero. Denmark lost to the UK in the final game. He was such an accomplished football player that it is said when he defended his PhD thesis there were more football fans in the audience than mathematicians.
Brother Niels was also a good athlete, but often seemed to have his focus somewhere other than sports. Both brothers played several games for the Copenhagen-based Akademisk Boldklub, with Niels in goal. The story is told that during one game when almost all the action was happening in the attacking half for his club, a long clearing kick from the other end of the field began to roll toward his goal. Niels stood near the goalpost and seemed unaware of the ball rolling toward his goal with players rushing in from many yards behind it. Alerted by the screaming crown behind him, Niels made the save and cleared away the threat.
After the game his explanation was that he had been distracted by a math problem and was carrying out calculations on the edge of the goal post.
Apparently he kept his love for the game.  The photo at the top shows Niels Bohr with  a group that is unidentified, but he looks to me to be the one handling the ball.  A goal-keeper to the end, it seems.

An Anon. comment suggested, "I think there's also Gamow (upright), Pauli (back to camera) and possibly Heisenberg opposite Bohr. Must have been during a conference."  If anyone can spot brother Harald, and if someone recognizes young version of later great science/math wizards, share, please

From Surds, to Ab-Surds

I still use the word surd for irrational square roots, and I know there is a undercurrent in modern math education to remove what is considered "difficult"  language in the classroom.  I leave that to those still fighting in the classroom to decide, but as a historian, the term is too rich in content not to use it, and teach it. Hence the title of From Surds, to Ab-Surds


When I first saw the image above I thought, Oh, that's neat. I mean I know it doesn't normally work, but then I also like the crazy wrong cancellations that work, such as

If you restrict yourself to two digit numbers, there are three more of these.  For folks who want to search them out, I will give the four at the bottom of the post.  

And in case you wondered, there are three digit examples also, and onward.  Here are a couple to get you started

\( \frac{106}{625}\),  ... \( \frac{116}{464}\),  ... and for variety \( \frac{98}{392}\),  ... 

 I know you were wondering, and yes,  it goes on... 

\( \frac{1019}{5095}\),  ... 

  I know there are lots more, so if you expand on this list, send me a note.  

As I sat and tried to think of other similar "wrong" examples that work with surds, I realized it might make a really good first or second year algebra challenge.  There is nothing very difficult about the algebra itself, so it allows the problem to be setting up the algebraic structure of the arithmetic problem.  

My early thoughts quickly generated enough to recognize a pattern to generate as many as I would want, *** 

 \( \sqrt {3 \frac{3}{8}} = 3 \sqrt{ \frac{3}{8}} \) 

*** or one in higher values.  For example:

 \( \sqrt {49 \frac{49}{2400}} = 49 \sqrt{ \frac{49}{2400}} \) 

  And in general it will always work in this form::

    \( \sqrt {n \frac{n}{n^2-1}} = n \sqrt{ \frac{n}{n^2-1}} \) 

Are there other patterns that would produce fractional oddities like these?  (Send them to me.)

I was reminded by Subramanian R that there is an easy extension of these to higher powers and roots .  For instance, the first problem can be adjusted to cube roots by using 2^3 -1 in the denominator.. 

 \(\sqrt[3](2 \frac{2}{7}) = 2 \sqrt[3](\frac{2}{7}) \)

 \(\sqrt[4](2 \frac{2}{15}) = 2 \sqrt[4](\frac{2}{15}) \)

And in general, \(\sqrt[r](k\frac{k}{k^r-1})= k\sqrt[r](\frac{k}{k^r-1})\)

********************************

The four two-digit false cancellations are 

\(\frac{16}{64}\)

\(\frac{19}{95}\)

\(\frac{26}{65}\)

\(\frac{49}{98}\)

An Unusual Periodic Table

Lemniscate The word lemniscate comes from the Greek word lemniskus for ribbon. The mathematical curve, a sort of figure eight, does look somewhat like the bow for a package made from a twisted ribbon [see figure]. The word is beginning to disappear from textbooks, and is completely missing in my high school edition of the American Heritage Dictionary. The only closely related term I could find was lemniscus, a term for a nerve bundle in the brain. No picture was available, but it may be this also looks like a ribbon.
The Mathematical curve [formulas below]is related to the rectangular hyperbola through the following relationship. If a tangent is drawn to the hyperbola and the perpendicular to the tangent is drawn through the origin, the point where the perpendicular meets the tangent is on the lemniscate.

I recently saw a picture of a chemical periodic table in the shape of a lemniscate created by William Crookes in 1888. The picture is on page 107 of The Ingredients: A Guided Tour of the Elements by Philip Ball.

Theophilus Grew, First Math Prof at Univ of Pennsylvania

  

December 17, 1750 - Mr. Theophilus Grew appointed first Master in Mathematics at Academy of Philidelphia (to become the Univ of Pennsylvania). Grew published the first American Trigonometry book while there, “The Description and Use of the Globes..” His 1752 Barbados almanack, for the year of our Lord 1752, being bissextile, or leap-year. / By Theophilus Grew, professor of the mathematics was published in 1751 and printed by Ben Franklin. "This is the only recorded sheet almanac extant from the Franklin shop and the only one prepared by Grew which Franklin and Hall are known to have printed."--*C. W. Miller, Franklin

No documentation survives of Theophilus Grew's birth, education and early life. He first appears in the historical record in the early 1730s, when he had the astronomical skills to calculate and prepare almanacs, probably in Maryland. His first known almanac, The Maryland Almanack for the Year...1733, was published in Annapolis in 1732. Later his almanacs were published in Philadelphia, New York and Willliamsburg.

 
The earliest extant newspaper advertisement identifying Grew as a schoolteacher appeared in Philadelphia in 1734. In his Philadelphia school he taught mathematical subjects, including basic arithmetic as well as such academic and practical applications as surveying, navigation, astronomy, accounting and the use of globes. After serving as a headmaster in Chestertown, Kent County, Maryland, from 1740 to 1742, he reopened his school in Philadelphia. 

This school for boys, located first on Walnut Street and then in 1744 in Norris Alley, included English, but emphasized mathematics. It also included a night school offering mathematics for gentlemen. In 1739 Franklin's nephew James Franklin, Jr., and his son William Franklin had been studying with Theophilus Grew, but on 12 December, they became students of Alexander Annand.
A friend of Benjamin Franklin.

 Grew was one of Penn's earliest faculty members. He was appointed master of mathematics at the Academy of Philadelphia (origin of the University of Pennsylvania) in 1750. When the College charter was obtained in 1755, he was elected the college's first professor of mathematics. He was mathematics professor for Penn's first graduates, the Class of 1757. In 1753, the trustees gave Grew permission to conduct a private evening school as well. The curriculum of one of Grew's courses is revealed in a notebook kept by John Yeates Jr. while a student at the Academy in 1753; the notebook is entitled “An Introduction to the Mathematics: Containing Compends of Arithmetic, Geometry and plain Trigonometry, The different Kinds of Sailing, with The common Method of keeping a Journal at Sea.”

Besides his teaching, Grew also in 1749 published a description of an approaching eclipse of the sun, and the following year served as one of the commissioners from Pennsylvania establishing the boundary between Pennsylvania and Maryland. Grew's most important publication was his student text, The Description and Use of the Globes, Celestial and Terrestial; with Variety of Examples for the Learner's Exercise (a copy of which was presented to the University Library in 1905). This 1753 book was the first textbook by a member of the Penn faculty as well as first student text on the use of globes to be published in the American colonies. 

For his contributions to knowledge, he was awarded the honorary Master of Arts degree at the 1757 commencement. Grew entered his son Theophilus in the Academy in 1751. He served as Penn's mathematics professor until his death from consumption in 1759. He is buried in The Christschurch Burial Ground with many notables including Ben Franklin, and five other signers of the Declaration of Independence. *archives.upenn.edu

White Rabbit Math - Extended

 

***I first wrote most of this post in 2008, but then an event reminded me of it, and so I thought I would add on to this old, but still interesting post with an additional interesting connection.

One of the things that amazes me, and I think most people who are attracted to math, is the mysterious way that different parts of math come together in unexpected ways. I tried to explain this to someone once using a literary analogy..."It is as if you were reading along in some great drama, or trying to understand the message in some grand poem, and suddenly the White Rabbit from Alice in Wonderland comes running through muttering, "Oh dear! Oh dear! I shall be too late!"
It is not the White Rabbit you see in math, but the effect is the same. Euler must have felt that feeling after he struggled to find the value of the series \( \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2}+ ...\).. and finds that it turns out to be \( \frac{\pi^2}{6} \). Wait.... Pi is the ratio of the circumference to the diameter of a circle, but there are no circles in the sum of the squares of the reciprocals of the integers; and yet, there it is, the mathematical white rabbit coming seemingly from nowhere. Certainly none of the many mathematicians of great repute who had worked on the problem found (or expected) Pi to appear.

The normal distribution is another example; De Moivre takes the binomial probability distribution for flipping a coin and generalizes it toward an infinite number of flips, and POW, the normal or bell-shaped curve that is ubiquitous in intro stats. And what happens? Right there in the middle, the height of the normal curve at Z=0 is .39894... No, NO, NO, NOT JUST .39894.. but the .39894... that is exactly equal to \( \frac{1}{\sqrt{2 \pi}} \)

Ok, so what brought this sudden rebirth of excitement about mathematical interrelationships? Well recently I came across a blog that referred to another blog that (as these things sometimes do) led me to a paper on just such a mathematical "white rabbit". The paper was about partitions of numbers as powers of two (1, 2, 4, 8, 16, etc..)
It began with a simple question, what is the number of ways to write a number n as a sum of powers of two if each value can be expressed no more than two times. For example, we could express 4 as 4, or as 2+2, or as 2 + 1 + 1 since each value is a power of two, and none appears more than twice. You couldn't use 1+1+1+1 since it appears more than twice. For n= 4 it turns out that the number of partitions, as shown above, is three. If we assume that there is one way to express zero, and one way to express one, and figure out the others we get a string like this


1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7,..
Ok, you don't see a white rabbit yet... but then someone ask you a different question. Is it possible to write out ALL the rational numbers in simplified form without repeating any of them. The answer is "Yes, of course, see the list above."
"What?", you ask, "How?", but there it is... The sequence of rational numbers is formed by taking each of the numbers to be the numerator, and using the number behind it to be the denominator. 1/1; 1/2; 2/1; 1/3; 3/2; ... and you never get a repeat, never get an unsimplified form, and you eventually get them ALL, the entire Infinite Set.....
No way you would expect that partitions of powers of two should give you the rational numbers in their entirety... there is (it would seem) nothing to relate the two questions... and yet... there it is. I think that is what makes math the most exciting area of study in the world.
Prove it you say? Nope, In truth I ain't man enough, but you can find the entire paper
Recounting the rationals, by Neil Calkin and Herb Wilf. Read their proof and Enjoy.

*** So today I was catching up on some old audio podcasts from "My Favorite Theorem," and Jordan Ellenberg   was explaining his choice of a special part of Fermat's Little Theorem, that for any prime p, \( 2^p \equiv 2 Mod p \).   (or in very primitive terms, if you divide 2^p by p, you always get a remainder of 2.  I wondered why he found that so interesting, but then he hit me with, "you can discover at least that it’s true on your own, for instance by messing with Pascal’s Triangle, for example." And of course, in a moment I realized yes, Fermat's Little Theorem, at least this limited case, is elementary true by looking at the rows of Pascal's Triangle. The sum of all the elements of any row add up to a power of two, and the pth row has a sum of 2^p. But look at some prime row.....

the 3rd has 1,3,3,1 ;

the fifth has 1,5,10,10,5, 1 ;

and the 7th has 1, 7, 21, 35, 35, 21, 7, 1....

In each row, all the entries are divisible by p, except the two ones. Scan the rest and you notice the same thing. And just importantly, you don't have to go very far to see an exception for the non-primes.

Math has those White Rabbits everywhere.

"Holy Cow", Holy Water, Heron Invents the Vending Machine

 

Reposted from 2011:

When my algebra students were first introduced to Heron's (or Hero's) formula, I always told them a brief historical note about his invention of a steam-jet propelled automaton (called the æolipyle) that he created during the first century of the common era. It was just a novelty experiment and didn't do a lot, but let's put it in some historical perspective. Once Hero's aeolipyle was forgotten, we don't know of any other person inventing a steam engine until the Ottoman inventor and all-around genius Taqi al-Din in 1577 - and he was considered the greatest scientist on Earth by his contemporaries.

Over Christmas I received the little "Book of Secrets" as a gift and learned as I leafed through it that he was also the inventor of the first known vending machine. Apparently in ancient times folks were required to pay for holy water to wash themselves before entering the temples, but it seems they didn't always cough up the cash..... so... Heron invented a device to help keep them honest.

http://kotaku.com

Here is how the device operated as provided on the Smithsonian Museum web page:

How it works: A person puts a coin in a slot at the top of a box. The coin hits a metal lever, like a balance beam. On the other end of the beam is a string tied to a plug that stops a container of liquid. As the beam tilts from the weight of the coin, the string lifts the plug and dispenses the desired drink until the coin drops off the beam.
Proof of complexity: Early modern vending machines actually used a similar system, before electrical machines took over.

According to Erik Davis's book "Techgnosis: Myth, Magic & Mysticism In The Age Of Information", Heron actually designed robots to perform entire plays. They would move about the stage, enter and exit on their own, and

Another staged a Dionysian mystery rite with Apollonian precision: Flames lept, thunder crashed, and miniature female Bacchantes whirled madly around the wine god on a pulley-driven turntable.

And just so you fully experienced the drama, it was complete with sound effects. Here is the machine he used to produce the effect of thunder. the Yeah, I would be lined up to see that show myself.

mlahanas

I much more recently found a site with a schematic of Heron's water fountain.  The two chambers B and C must be airtight, thus allowing the lower dish to siphon up to the upper.  More technical matter can be found here.  

He also invented a wind wheel operating a pipe organ—the first recorded instance of wind powering a machine .

"More illustrated technical treatises by Heron survived than those of any other writer from the ancient world. His Pneumatica, which described a series of apparatus for natural magic or parlor magic, was definitely the most widely read of his works during the Middle Ages; more than 100 manuscripts of it survived. However, the earliest surviving copy of this text, Codex Gr. 516 in the Bibliotheca Marciana in Venice, dates from about the thirteenth century— a later date than one might expect. Conversely, the complete text of Heron's other widely known work, the Mechanica, survived through only a single Arabic translation made by Kosta ben Luka between 862 and 866 CE. This manuscript is preserved in Leiden University Library."  *HistoryofInformation.com

 

How the term Scientist came to be-

 

Re-posted from a 2011 Post:

John Cook, at the Endeavour just wrote a nice tid-bit about science/language regarding the creation of the word scientist. I knew the story, but apparently from a flawed source as I had credited the wrong poet (I had Wordsworth... not a bad poet, but not correct) .. so I will correct my notes, and along the way, supplement John's blog with a little more interesting detail from my notes about the topic.. (I think most of this is right).
John Wrote:

For most of history, scientists have been called natural philosophers. You might expect that scientist gradually and imperceptibly replaced natural philosopher over time. Surprisingly, it’s possible pinpoint exactly when and where the term scientist was born.

It was June 24, 1833 at a meeting of the British Association for the Advancement of Science. Romantic poet Samuel Taylor Coleridge was in attendance. (He had previously written about the scientific method.) Coleridge declared that although he was a true philosopher, the term philosopher should not be applied to the association’s members. William Whewell responded by coining the word scientist on the spot. He suggested

by analogy with artist, we may form scientist.

Since those who practice art are called artists, those who practice science should be called scientists.

This story comes from the prologue of Laura Snyder’s new book The Philosophical Breakfast Club. The subtitle is “Four Remarkable Friends Who Transformed Science and Changed the World.” William Whewell was one of these four friends. The others were John Herschel, Richard Jones, and Charles Babbage.

I also found a note about what might be the first use in print.  "Interestingly, the philosopher William Whewell wrote a review of On the Connexion of the Sciences in 1834, ( by Mary Sommerville) and in the review he coined the word "scientist" as an appropriate name for a person who dabbles in experimental natural philosophy. "(*Linda Hall Org) 

John Cook was good enough to share some material from the book and I would strongly recommend it to anyone who enjoys the history of science, or just a good story. This is my first read of anything by Laura Snyder, but I hope it will not be the last.

And now, here is the part from my notes which I hope adds to the story:

Whewell was also frequently in correspondence with Michael Faraday, and created the scientific terms anode, cathode, and ion. A letter between the two discussing these three terms is in the Wren Library at Trinity College in Cambridge. I have tried to capture an image below, but the library does not allow flash and the image is taken through the glass case... my apologies that it is not clearer.

In spite of its creation at such a high academic level, the word scientist was not well accepted for a long time. Its eventual acceptance came first in America, but it seems even there it encountered fierce opposition to its formal use well into the Twentieth Century. In The American Language in 1921, H. L. Mencken wrote

The last-named scientist was coined by William Whewell, an Englishman, in 1840, but was first adopted in America. Despite the fact that Fitzedward Hall and other eminent philologists used it. Despite this fact an academic and ineffective opposition to it still goes on. On the Style Sheet of the Century Magazine it is listed among the "words and phrases to be avoided." It was prohibited by the famous Index Expurgatorius prepared by William Cullen Bryant for the New York Evening Post, and his prohibition is still theoretically in force, but the word is now actually permitted by the Post. The Chicago Daily News Style Book, dated July 1, 1908, also bans it. The use of the word aroused almost incredible opposition in England. So recently as 1890 it was denounced by the London Daily News as "an ignoble Americanism," and according to William Archer it was finally accepted by the English only "at the point of the bayonet."

The term Natural Philosopher which scientist replaced had not been around long itself. Prior to the time of Galileo a Philosopher was indifferent to the observed facts, and dealt only with moral and logical theory. Galileo thought that,"The proper object of Philosophy is the great book of nature..." and not the words of other men. Eventually these new students of the "book of nature" became the "Natural Philosophers".

Despite several common assertions to the fact that Whewell coined the term in 1840,[Did they get the wrong date?... see date above in John Cook's story] the OED lists an earlier use in print, "1834 Q. Rev. LI. 59 Science..loses all traces of unity. A curious illustration of this result may be observed in the want of any name by which we can designate the students of the knowledge of the material world collectively. We are informed that this difficulty was felt very oppressively by the members of the British Association for the Advancement of Science, at their meetings..in the last three summers... Philosophers was felt to be too wide and too lofty a term,..; savans was rather assuming,..; some ingenious gentleman proposed that, by analogy with artist, they might form scientist, and added that there could be no scruple in making free with this termination when we have such words as sciolist(***see below), economist, and atheist but this was not generally palatable."

It seems like they are talking about the same event with a different date on this article.

William Whewell is buried in Trinity College Chapel in Cambridge, UK. A memorial marker in the chapel is shown here and there is a statue in the ante-chapel

Addendum:  I recently came across notes that suggest that Faraday didn't really accept the term despite his close relation with Whewell and his public endorsement of it; "As for hailing [the new term] scientist as 'good', that was mere politeness: Faraday never used the word, describing himself as a natural philosopher to the end of his career."     It also appears he didn't like physicist, "[The new term] Physicist is both to my mouth and ears so awkward that I think I shall never use it. The equivalent of three separate sounds of i in one word is too much."  *Sydney Ross Nineteenth-Century Attitudes: Men of Science (1991), 10. 

A few years after I wrote this, Thony Christie featured a guest post on his Renaissance Mathematicus by  Dr Melinda Baldwin with lots more information.  The entire article is worthy of your reading, but I have "liberated" a few sections here:

"Most nineteenth-century scientific researchers in Great Britain, however, preferred another term: “man of science.” The analogue for this term was not “artist,” but “man of letters”—a figure who attracted great intellectual respect in nineteenth-century Britain. “Man of science,” of course, also had the benefit of being gendered, clearly conveying that science was a respectable intellectual endeavor pursued only by the more serious and intelligent sex."

And this which has lots of information about the 20th century use of  "scientist" even by "scientific" journals:  

"Feelings against “scientist” in Britain endured well into the twentieth century. In 1924, “scientist” once again became the topic of discussion in a periodical, this time in the influential specialist weekly Nature. In November, the physicist Norman Campbell sent a Letter to the Editor of Nature asking him to reconsider the journal’s policy of avoiding “scientist.” He admitted that the word had once been problematic; it had been coined at a time “when scientists were in some trouble about their style” and “were accused, with some truth, of being slovenly.” Campbell argued, however, that such questions of “style” were no longer a concern—the scientist had now secured social respect. Furthermore, said Campbell, the alternatives were old-fashioned; indeed, “man of science” was outright offensive to the increasing number of women in science.

In response, Nature’s editor, Sir Richard Gregory, decided to follow in Carrington’s footsteps. He solicited opinions from linguists and scientific researchers about whether Nature should use “scientist.” The word received more support in 1924 than it had thirty years earlier. Many researchers wrote in to say that “scientist” was a normal and useful word that was now ensconced in the English lexicon, and that Nature should use it.

However, many researchers still rejected “scientist.” Sir D’Arcy Wentworth Thompson, a zoologist, argued that “scientist” was a tainted term used “by people who have no great respect either for science or the ‘scientist.’” The eminent naturalist E. Ray Lankester protested that any “Barney Bunkum” might be able to lay claim to such a vague title. “I think we must be content to be anatomists, zoologists, geologists, electricians, engineers, mathematicians, naturalists,” he argued. “‘Scientist’ has acquired—perhaps unjustly—the significance of a charlatan’s device.”

In the end, Gregory decided that Nature would not forbid authors from using “scientist,” but that the journal’s staff would continue to avoid the word."  

---

*** Sciolist..... If you recognized this term you are ahead of me...I looked it up and found:

Noun

1.

sciolist - an amateur who engages in an activity without serious intentions and who pretends to have knowledge a dabbler,  a dilettante  (thank goodness they didn't use my name or picture) [From Late Latin sciolus, smatterer, diminutive of Latin scius, knowing, from scre, to know; (This is the same root that gives us science.)

More on a Geome-Treat with a Calculus Twist

I recently re-posted a blog I wrote 12 years ago about a way to find the tangent to the curve of a conic without employing calculus that was one of my favorite math "tricks". In response I got a serious question from Brandon@_nilradical who asked, "anything "similar" for cubics, quartics, etc?". Being busy and wanting to reply I passed off a hasty, "Nothing quite so simple." and went back to mowing. Later I felt guilty about dismissing the question, so I thought I would fess up to what little I have found in playing with the idea "Beyond the Quadratics."
(If you are not familiar with the point substitution approach to finding tangents, take a moment to read the older post, it's pretty brief.)

My first excursion was to try a simple cubic and see if I could figure out how to apply the same polar idea. What could be easier than y=x³ so I picked the point (1,1) and set out exploring. I first had to decide how to replace the three x's in the right side, and decided that I would replace y= x³ with (y+1)/2 =(x)(1)(x+1)/2

Ok, we got a tangent, but it was a tangent quadratic, not a tangent line. So I decided to press on and apply the idea recursively into the new parabola.  The parabola simplified to y=x² +x - 1, so I began to substitute into that using the point (1,1) again.  This leads to (y+1)/2 = (1)(x) + (x+1)/2-1

I didn't even bother to simply, just entered into the Desmos calculator and ....Eureka!!!

So how could we extract this and explain with a little calculus?  Well, if we begin by saying we want to create a parabola at the point (1,1) with the same slope (3) as the cubic there, we would begin with y=x² + bx +c  and since the derivative of that, y'=2x+b must equal 3, setting 3=2(1)+b we see that b must be 1 also.  Now we just plug (1,1) into the equation for y=x² + x +c and we quickly find that indeed, the calculus will give us y=x² +x - 1 as the parabola tangent to the point (1,1) with the same slope as y=x³

At this point I had no idea what this descending cascade of polar approaches to a tangent would do with something really complicated, but I barged ahead and created something minorly absurd.
So I started typing into the calculator creating as I go along and came up with x³y = y² +x+x<sup>4

Desmos responds with :</sup>

Ok, this looks like fun.  The first problem is finding a nice point with integers and a non-zero slope.... and (-1,0) jumps out because it should eliminate some congestion substituting y=0 in some places.

I begin by the same approach of using equal parts of variable and constant wherever possible, and write out (-1)(x)(x+1)/2 (y+0)/2= 0y +(x-1)/2 + (-1)(-1)(x)(x)  and we get a cubic with two infinite discontinuities, but one of the branches slashes through the point we seek

At this point I'm convinced that our descending iteration of polars will proceed to a line tangent at the same point... and I found it interesting that even picking the equation out of my head, the point I chose also had a slope of 3 at that point.

I'm not sure I have any idea how useful these techniques might be for those forging father than my simple experiences in math can anticipate, but if you are one who knows more about this, share what you know.

I posted tongue-in-cheek at the end of the first post that the challenge of proving that it always worked was a homework assignment for calculus students, due by Wednesday.  In response, Thomas Morgan replied:  "I can verify that this will work in general. Multivariable calculus tells us that to compute the tangent vector at a point, one need only compute the partial derivatives at that point. It should be clear from the method description that the new curve intersects the old curve at the point under consideration. Two different curves that intersect at a point are tangent at that point if their tangent vector points in the same direction, so we can scale our tangent if necessary. Using partial derivatives, in addition to the sum rule for derivatives, reduces the effort to computing single-variable derivatives for two cases. In the first case, consider the tangent of c*x^(2n) at the point x=t. Computing the derivative directly yields 2n*c*x^(2n-1). Using your method, we first transform c*x^(2n) to c*x^n*t^n. Its derivative is n*c*x^(n-1)*t^n, which at the point x=t only differs from the first calculation by a factor of 2. We similarly compute the derivative of c*x^(2n+1) at the point x=t. Direct calculation yields (2n+1)*c*x^(2n). Using your method, we first transform c*x^(2n+1) into (c*x^(n+1)*t^n + c*x^n*t^(n+1))/2. Its derivative is ((n+1)*c*x^n*t^n + n*c*x^(n-1)*t^(n+1))/2, which at the point x=t only differs from the first calculation by a factor of 2. Thus we see that the tangent vector to the curve given by your transformation is the same vector as the tangent vector of the original curve, compressed by a factor of 2. This shows that the new curve is tangent to the original curve, and by induction, we are done."  

Thank you for the response Thomas.  

Victorian Political Correctness, Math Terminology, and Urban Legends

 The year 1913 seems to have had a strange effect on educational language, and as yet, I haven't figured out exactly what happened.

A few days ago, Dave Renfro, an internet associate who does more research into journals than anyone I have ever heard of, sent me a note that had an aside that said, "Also, ...,I've seen the terms "promiscuous exercises" and "promiscuous problems".

I did a little follow-up and found literally dozens of books that use the phrase "promiscuous problems". My Google Book search on the exact phrase produced 71 books and journals, mostly referring to mathematics, but not exclusively. In glancing at the dates, I noticed that almost all were before 1900. So I set the same search with a cut-off of before 1900. The result?... There were still 25, but only five of them were after 1910. Of these five, one was about sexual disorders of bulimic patients and had nothing to do with problem sets of the educational sort, one was a catalog of antiquarian objects and was referencing a phrase in an older object, two were reproductions of very old texts. That leaves the one final object after 1910 that referred to Promiscuous exercises in regard to problem sets, with a date of 1911, John Henry Diebel's  Arithmetic by Analysis. For some reason, the usage to describe a set of problems or exercises seems to have disappeared after that date almost completely.

So what do they mean, "promiscuous" problems. One of the definitions leads back to the old Latin root. Here is the way they gave the etymology in the Online Etymology Dictionary:

"consisting of a disorderly mixture of people or things," from L. promiscuus "mixed, indiscriminate," from pro- "forward" + miscere "to mix" (see mix). Meaning "indiscriminate in sexual relations" first recorded 1900, from promiscuity (1849, "indiscriminate mixture;" sexual sense 1865), from Fr. promiscuité, from L. promiscuus.

So the term was essentially used for a general mixture, thus promiscuous exercises were a mixed review; but then in 1900 the phrase became associated with "indiscriminate in sexual relations" and apparently that usage became so common, that the use of promiscuous exercises was no longer classroom acceptable.

Makes me think of a story that John H Conway, told (I believe) about the word hexagon. If you search the word "sexagon" you will see that it was very common in old math texts, then during the Victorian era, it became too suggestive for classroom use, and so hexagon, which also has a long history of use, became the preferred term. The earliest use of Sexagon in English , according to the OED, was by A. Rathborne in Surveyor, written in 1616. The term in English probably came from the use of Latin as the language of choice in science.  Sex was the prefix for six ans still remains in words like sextant, sexagenarian, and sextet.  Its demise may have been due to the hybrid nature of the word, sex from Latin and gon from the Greek for knee.  Hexagon was a union of two 

After I wrote this I got a comment:

r.r. vlorbik said...

i've heard, though never verified, that Victorian prudery also caused certain teachers to begin referring to the "arms" rather than the "legs" of a right triangle (the non-hypotenuse sides).
this is to say nothing of "parent function".i've *always* thought of the higher node
of a link in a tree as the "parent' of the lower... so this terminology (in discussion of
function transformations; x^2 is the "parent" for 3(x-2)^2 +1... you know the drill...) seems perfectly natural to me.

but somebody with some public-school experience told me,what may even be true, that up to a point,
one had called these things *mother* functions. which had to be made to stop.... latus rectum. (wrecked 'em, hell... it killed 'em).

So I was off on another search:

I have a pretty extensive collection of old textbooks, including many British texts, and I didn't remember ever seeing anyone use "arms" in that fashion, so my first thought was that, if it were true, it was only a very minor usage. Since the good lady Victoria, ruled from 1819 to 1901, I thought I would search before and after her reign.

I pulled out my 1804 edition of Playfair's "Elements of Geometry", published in Edinburgh. He referred to the right triangles sides as..."sides"... His Book VI, prop. XXXI reads exactly like the Thomas Heath Translation. No help there, so I skipped forward 99 years to the other end of the Victorian period, 1903 and looked in "A Junior Geometry" by Noel S Lyndon, published in London, only to find he also used only the terms hypotenuse and "other two sides" in his statement of the Pythagorean Thm.

Perhaps neither term was common in the Victorian period, and these stories were a bit of urban legend. I went on to Google Books to see if I could find any examples of geometric usage such as Vlorbik had described..... I entered a search for "arms 'right triangle' geometry"

....Yikes", there they were. The first listing was "Plane Geometry" by Arthur Schultze, Frank Louis Sevenoak, Limond C. Stone, from 1901. It contained, "The sum of the squares of the arms of a right triangle is equal to ..." along with 388 other listings, some dated as late as 2008. "In a right triangle whose arms have lengths a and 6, find the length of the .." appears on page 451 of the fourth edition of Schaum's Outline of Geometry from that year.

 Ok, but that still did not mean it was the influence of the dreaded Victorian stuffed-shirts... I switched the cut-off to 1850... and there were NO results prior to that year... only one last check. Would there be examples with the use of "legs" prior to that year? There were indeed, including several by the famous American Mathematician, Benjamin Pierce. Another from 1734 was from the British Benjamin Martin.

So it appears that there was some pressure to use "arms of a right triangle" suggested by these dates; but there is still no smoking gun. One observation that suggests that if such a suppression existed, it may have been much more influential in the US than in England. One is that the OED gives no reference to the use of arm as a mathematical or geometrical term.  The other is that most of the books  found using "arm" seemed to be of US origin. Does anyone out there know of a document or statement of any kind in the math education literature that makes a clear suggestion to teachers? If you know of such a document, please share whatever level of information you have and I will pursue it.

Functions, Parents, and Parent Functions

From what I have been able to discover in a short period of research, the use of terms like "parent function" seems to have worked its way into mathematics from statistics, which seems to have gotten it from the anthropologists/sociologists.

Across America and Asia: Notes of a Five Years' Journey Around the World ...‎ - Page 250 by Raphael Pumpelly - Voyages around the world - 1870
"if it should bear the same relation to the parent population that.."

Prior to 1900 there are almost no listings of "parent function" in a mathematical usage. Around the end of the 19th century, statisticians began to talk about the distribution from which a sample was taken as the "parent distribution" of the sampling distribution (population of all samples of some size n). Some of these may be related to the study of eugenics in which the study was about the relations of some characteristic of the offspring to the actual parents, but the usage grew.

Data reduction and error analysis for the physical sciences , by Philip R. Bevington - 1969 10-2 the F TEST As discussed in the previous section, the x^2(chi square) test is somewhat ambiguous unless the form of the parent function is known because the statistic
x^2 (chi-sq)... "

Occasionally I find the term "parent function" applied in this way when the distribution of the original sample was a normal distribution.
The Annals of mathematical statistics‎ - Page 179
by American Statistical Association, Institute of Mathematical Statistics, JSTOR (Organization) - 1948" Usually the parent function is the Type A or normal curve, as discussed by Gram "

There are also some early uses of the "parent function" in association with the use of inverses and derivatives in calculus and analysis texts back to about 1925. By 1970 the term had become commonly understood, but not abundantly used.

The teaching of secondary mathematics‎ - Page 521
by Charles Henry Butler, Frank Lynwood Wren - Education - 1965 - 613 pages
".. of an inverse function and its relation to the parent function or else in failure to attach clear meanings to the terminology and notation employed. ..."

It was the 1980's and the introduction of computers and graphing calculators into modern classrooms that seemed to make the term "parent function" ubiquitous. Any function that appeared on the calculator was a parent function, and the translations, rotations, shears, etc became the "children".
I think this use also led to the introduction of "mother function" rather than the other way around. I can only find a few examples of "mother function" and there does not seem to be any pattern to the frequency as one might expect if a term had arisen to replace this one as an "off-color" predecessor. In fact, it seems "mother function" is more commonly used by continental writers, often in conjunction with "daughter functions"; but admittedly the sample size I have to draw on was small. Perhaps these were early pioneers for language equality.

If you are one of those people with access to old journals, or a collection of old texts, I would appreciate any references to the use of any of these terms and a source earlier than 1900; and if you have a way to make a digital copy and send it by email, I will have my students name their children after you.