Colossal Gardner, ch. 7

The next article for Plane Geometry is on Penrose Tiling.

Unlike the earlier discussion of Rep-Tiles, which are periodic, Penrose Tilings are nonperiodic, and were first discovered by Roger Penrose. Periodic tilings can be made by shifting the pattern in specific directions without rotation or reflection. These are used extensively in the pictures by M. C. Escher. Nonperiodic tilings, at a minimum then, involve rotating or mirror reflecting the tiles. A very simple nonperiodic tile is the triangle. put two triangles back to back to form either a larger triangle, or a 4-sided rhomboid; the tiles themselves get rotated or flipped, but the patterns they make may themselves be periodic. Periodic tiles can be made nonperiodic by giving the edges different colors and requiring that only tiles with matching colors can be adjacent to each other.

Gardner laments that Escher died before learning about Penrose tiles, but I think M.C.’s butterflies print qualifies. Which is interesting to me, because Roger’s father, geneticist L.S. Penrose, invented the unending Penrose Staircase, which Escher depicted in his “Ascending and Descending” woodblock print. (Roger himself worked in general relativity and quantum mechanics.) In 1973, Roger found 6 tiles that are nonperiodic, with notches and tabs that forced the tiles to only fit one way. That was lowered to two triangular tiles, called “kites” and “darts”. Before making them public, he filed patents in the U.K., U.S. and Japan. You can now buy them as a game from Kadon Enterprises.

Penrose tiling, from the wiki entry. (All rights belong to their owners. Images used here for review purposes only.)

In 1993, John Conway came up with a 3D object that could be used to tile an enclosed volume. Called a “biprism”, these objects stack on each other, but in an irregular pattern. Currently referred to as a Schmitt-Conway-Danzer biprism, the cut-out pattern shown below was created for Gardner by Doris Schattschneider, co-creator of the M. C. Escher Kaleidocycles, and first female editor of Mathematics Magazine.

An example of tiling Penrose chickens (from the Martin Gardner book)

The Penrose Tiles are tied to the discovery that quasicrystals, crystals with five-fold symmetry, are possible.

Make your own Conway biprism

I tried to make 4 biprisms, and all four failed in exactly the same way. Granted, I was using very flimsy copy paper, but the final fold when I glued the arms together, always pointed at least 5 degrees off-center, making me think that maybe there’s something wrong with the original pattern. It might be better to use thicker paper stock, and then cut the sides into separate pieces and hold then together with cloth tape. Anyway, the instructions to make these are – use a dried-out pallpoint pen to score along all the lines. Then fold the arms marked “u” upward (valley folds) so that those arms pair up to make a prism on the top of the central rhomboid. And, fold the arms marked “d” downward (mountain folds) to have a matching biprism on the bottom side of the rhomboid.

You may have more fun playing with the Kadon Enterprises game sets, or with your own set of Penrose chickens.

Wednesday Answer

Solution: What is the smallest convex area in which a line segment of length 1 can be rotated 360 degrees? (A convex figure is one in which a straight line, joining any two of its points, lies entirely on the figure. Examples include circles and squares.)

Answer, an equilateral triangle of height 1. For the line segment to rotate, the sides have to be at least length 1. Of all convex figures with widths of 1, the equilateral triangle has the smallest area. Try taking a toothpick and a cardboard cutout of a triangle and check for yourself. To make it easier, glue a second toothpick in the middle of the first one to make an axle for rotating the first toothpick within the triangle cutout.

Colossal Gardner, ch. 4

We now move into Plane Geometry and the concept of Curves of Constant Width. Actually, the main content of Gardner’s article is captured in the wiki entry. His starting point is that you can use Reuleaux triangles in the place of wheels in a roller-based system, and a platform placed on top of the rollers will remain level and steady as the rollers rotate across a flat surface. So, it’s not necessary to use circular wheels if you don’t want to. (If you want pictures, go to the wiki article.)

The more interesting application of the Reuleaux triangle was in the Watts Brothers Tool Works patent for a drill and chuck system capable of drilling square holes. They also manufacture drill bits for pentagonal and hexagonal holes that have sharper corners. Although the outer edge of a curve of constant width is in contact with its bounding space at all times (as with a square for the Reuleaux triangle), the center of rotation moves around all over the place. That means that for the Watts square-hole bit, a special chuck is required to trace out the correct rotation to ensure that the hole actually comes out square. You can see the patents (filed in 1917) at Google patents: US1241175, US1241176 and US1241177.

The puzzle this time is based on the Kakeya Needle problem. What is the smallest convex area in which a line segment of length 1 can be rotated 360 degrees? (A convex figure is one in which a straight line, joining any two of its points, lies entirely on the figure. Examples include circles and squares.) Check the wiki article on Kakeya sets for an illustration of the Kakeya Needle.


Wednesday answer

Wednesday answer:
Find a copy of J. A. Lindon’s “Doppelganger“.


Colossal Gardner, ch. 3

The last chapter of the arithmetic and algebra section is on palindromes. I absolutely love the “near miss” palindrome at the front of the article, attributed to Ethel Merperson, in Son of the Giant Sea Tortoise, edited by Mary Ann Madden (Viking, 1975) –

A man, a plan, a canal – Suez!

Palindromes can take many forms, from words and sentences that can be read the same forward and backward, numbers that can be rotated, palindromic primes, and even photos of things (like a bird in flight, going from wing tip to wing tip).

Here’s a game. Start with any positive integer. Reverse it and add the two numbers together. There’s a conjecture that you’ll get a palindrome after a finite number of steps.


121  <— End



13431 <- End

There have been papers written on the existence of palindromic primes and powers. You can play with palindromic roots to get palindromic squares, such as 121^2 = 14641.

You can find many of the same language examples in the wiki article. Yreka City in California used to have the Yreka Bakery and Yrella Gallery. Then there was the former premier of Cambodia, Lon Nol. And, you can have sentences where the word order is palindromic: from J. A. Lindon – “You can cage a swallow, can’t you, but you can’t swallow a cage, can you?”

Note: Lindon was a pioneer in the recreational mathematics field of anti-magic squares. In magic squares, the rows and columns all add up to the same number. With anti-magic squares, the sums of the rows, columns and diagonals are all different. He died in 1979.

Challenge: Can you find a copy of J. A. Lindon’s “Doppelganger,” and can you write a longer palindromic poem yourself?

Comments: I love palindromes, and I had a book collection of them at one time. Back when I first had a 4-banger calculator, I’d tried to find palindromic numbers. I’ve never heard of anti-magic squares before, but they could be fun to play with.

Coconuts answer

There are actually two variants for the coconuts puzzle. In the older, easier one the monkey gets an extra coconut at the end. Versions of this one apparently go back to the middle ages. The newer version used by William, where the monkey only gets 5 coconuts, is more difficult to solve.

The Diophantine approach is to build up a polynomial as follows (for the older version of the puzzle):
N = 5A + 1
4A = 5B + 1
4B = 5C + 1
4C = 5D + 1
4D = 5E + 1
4E = 5F + 1

N is the original number of coconuts, and F is the number that each of the 5 sailors received at the end. The way to read this is, at the beginning, sailor A divided the pile “N” 5 ways, kept one pile of size “A”, and had 1 coconut left over that he gave to the monkey. Now, there is a pile of 4*A coconuts that is divided 5 ways by sailor B. He keeps one pile of size “B”, has one coconut left over for the monkey, and puts the other four B-sized piles back together again, etc.

If we turn this into one equation by normalizing A, B, C, D and E (i.e. – E = (5F+1)/4; D = (5E+1)/4 = (5F+1)*5/16; etc.) we get 1,024*N = 15,625*F + 11529.

We want the smallest positive value of N such that both N and F are integers. This is easy to do in VBScript as a simple program to run N from 1 to 10,000, but not so easy to derive by hand. (My solution is N = 15,621, F = 1,023.)

One of the interesting things about the Diophantine equations is that some of them have only one solution, others have infinite solutions, and others have none. In our case, the coconut problem has an infinite number of solutions, so we want the smallest one.

However, Gardner presents a twist approach using negative coconuts, which dates back to Norman Manning in 1912. Alternatively, you could take some of the coconuts and paint them blue to be easier to track. The reasoning goes – if you’re dividing up the coconuts into 5 equal piles 6 times, with no leftover coconut for the monkey at the end, then the smallest number that would work is 5^6 = 15625. Now, take four of those 15,625 coconuts, paint them blue and put them in the bushes. This leaves you with 15,621 coconuts, which you CAN divide into 5 piles, with one for the monkey for round 1. Put the four blue coconuts plus the other 4 piles together to make one pile of 5^5 coconuts. This can obviously be divided by 5. But, we pull the 4 blue ones and give the extra to the monkey. In Gardner’s words “This procedure – borrowing the blue coconuts only long enough to see that an even division into fifths can be made, then putting them aside again, is repeated at each division.” After the last round, the blue coconuts are left in the bushes, not belonging to anyone.

The use of blue, or negative, coconuts explains something I discovered when I ran my program. One of the solutions is for N = -4, and F = -1. Adding 5^6 (15,625) to this N gives 15,621, which is the next solution my program found, with f = 1,023. All subsequent solutions have the form N = k*5^6 – 4, and F = k*4^5 – 1 (where k runs from 1 to infinity).

The above description is for the variant of the puzzle where the monkey gets that 6th coconut in the final round of dividing the coconuts into 5 piles. For William’s version, where the last round doesn’t have the extra coconut, the Diophantine equation is:
1,024*N = 15,625*F + 8404

There’s a different general equation that can be used here, depending on whether “n”, the number of sailors, is even or odd:
# Coconuts = (1 + nk)*n^n – (n-1)  : for even n
# Coconuts = (n – 1 + nk)*n^n – (n-1) : for odd n

k is the multiplier used above to get infinite solutions as multiples of N, and for the lowest positive solution, k=0.
For n = 5 sailors,
N = # Coconuts = (1 + 0*5)*5^5 – (5-1)
N = 1*3125 – 4
N = 3,121
The last round of divvying will give 5 piles of 204 coconuts, and nothing extra for the monkey.

Finally, the last puzzle on Monday had three sailors finding the pile of coconuts. The first sailor takes 1/2 of the pile, plus half a coconut. The second sailor takes half of the remaining pile plus half a coconut. The third sailor does the same thing. Left over is exactly one coconut, which they give to the monkey. How many coconuts did they start with?
If you work backwards,
(1 + 1/2) * 2 = 3
(3 + 1/2) * 2 = 7
(7 + 1/2) * 2 = 15
Answer: 15 coconuts

Comments: Man, I never expected this chapter of the book to give me so many problems. I’d initially thought I’d spend an hour writing it up and that’d be the end of it. But, I kept making mistakes in forming the Diophantine equations for each puzzle, and both my math, and my VBScript program kept coming out wrong. I spent close to 2 days on this one.

Obviously, I’m not as good at recreational math puzzles as I’d liked to think I was.

The Colossal Book of Mathematics: Classic Puzzles, Paradoxes and Problems

(Image from the amazon page, used for review purposes only.)

The Colossal Book of Mathematics, by Martin Gardner (2001)
I’ve mentioned a couple times that I received this book for Christmas. It’s a big book (700+ pages, hence the name), so it’s taken time to read it all the way through. Gardner sold his first article to Scientific American in 1952. It was on logic machines, which were used to solve logic problems in the days before computers had come out. His article included a heavy paper insert with cut-out window cards he’d developed for solving syllogisms. The article was expanded and included in his Logic Machines and Diagrams book in 1959. It wasn’t until 1956, though, when he submitted an article on hexaflexagons (folded paper rings that you can turn inside out) that the SA editors were impressed enough to ask if he had enough material to start up a dedicated monthly column for them. “Mathematical Games” started on Jan., 1957, and ran until 1981, for almost 25 years and 300 columns. Colossal contains 50 of the best articles, hand-picked by Gardner, to cover subjects from topology and tiling to time and pseudo-science. About half the articles include puzzles you can try to solve, with answers at the end of each. All have addendums updating the columns with more recent discoveries, and some have reader response letters reprinted with corrections and alternative answers to the puzzles.

It’s a fun, challenging book. I skipped the puzzles on subjects I’m not that interested in, including the ones on topology about how to cut a doughnut to get two interlocking rings. Some of the other puzzles or projects are more time-consuming than what I want to tackle now, but…

I have nothing else to write about on this blog right now. There’s still nothing from Gakken regarding new kits, and none of the other science publishers has anything new out I want, either. So, I’m going to switch over to commenting on each of the articles in the “Colossal” book, on a weekly basis. Some of the comments may be short, just mentioning what subject that article covered, and others may have longer commentary or reprints of the puzzles (including photos of hexaflexagons). If you’re interested in this kind of thing, you may be better served by buying the book (definitely worth the $16 for a new copy). Otherwise, you could think of this series as a “52-week a year calendar” kind of thing, minus 2 weeks.

Ok, getting started. Any minute now. Here it comes. Ready… almost… just about now…

The first three articles are on Arithmetic and Algebra, with number one entitled “The Monkey and the Coconuts.” It centers on a short story written by Ben Ames Williams in the Oct. 9th, 1926 issue of The Saturday Evening Post, called “Coconuts.” In Williams’ story, there’s a simple puzzle that the plot hinges on. Five men and a monkey are shipwrecked on an island, and they spend the first day gathering coconuts for food. The coconuts were all piled up together and the men went to sleep. In the middle of the night, one of the men woke up, and decided to take his share right then. He divided the coconuts into 5 piles, with one left over that he gave to the monkey. He hid his portion, put the rest back into a pile, and went to sleep. Then, one by one, the other 4 men woke up and did the same thing, taking 1/5th of the coconuts, and always having one left over that got handed to the monkey. In the morning, they divided up the last of the remaining coconuts, and it came out to equal shares. Question: How many coconuts were there at the beginning (and by extension, how many were there at the end when they were handed out)?

Gardner lumps this puzzle in with a group of brainteasers called Diophantine equations, which are polynomials, usually with two or more unknowns, for which you want only the integer solutions. They are named after Diophantus of Alexandria, a 3rd century BCE Hellenistic mathematician who studied these equations, and was one of the first mathematicians to introduce symbolism to algebra. The approach Gardner uses is with negative coconuts, thus avoiding the need for a polynomial format. An easier variant of the problem is: Three sailors find a pile of coconuts. The first sailor takes 1/2 of the pile, plus half a coconut. The second sailor takes half of the remaining pile plus half a coconut. The third sailor does the same thing. Left over is exactly one coconut, which they give to the monkey. How many coconuts did they start with?

Answers on Wednesday.


Site problems – edited

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Gakken Kaeda Kit comments

Okay, the latest Gakken kit is finally out – the Kaeda drone, so named because the main prop blade resembles a maple seed. 3,980 yen ($39 USD) without the 8% tax. It’s been over a year since the last kit came out, in Sept., 2015, and the anticipation for the Kaeda Drone was probably blown out of proportion because of it. This one wasn’t much of a challenge to build, since the drone itself was already pre-assembled. The controller required assembly, but it only consisted of the two halves of the case shell, the battery cover, three knobs, the circuit board, and 6 screws, (and there are 2 replacement propeller blades).

(The controller parts, plus the two replacement blades.)

The suggested assembly time was 15 minutes, and I think I did it in 10. (It takes 4 AAA batteries.) The only issue was with the power LED leads, which had been bent 180 degrees, and the requirement is for the leads to be bent 90 degrees so the LED is sticking out the side of the case. But, that’s an easy fix. The drone is powered by a lithium polymer battery that takes about 30 minutes to recharge to 60% when plugged into the controller. That will give you roughly 7 minutes of flight time. If you want the battery at full power, you have to give it a second charge. The instructions are: 1) Turn off the controller and the drone. 2) Pull the charge cable out of the well at the back of the controller, under the battery cover. Plug the cable into the drone. 3) Turn on the controller power switch. The green LED will light on the controller. When it goes out, the first charge cycle is finished. 4) Repeat steps 1-3 for the second charge.

(The assembled controller.)

The controller talks to the drone via an infrared LED, so it has to be aimed directly at the drone at all times, or the drone will lose signal and touch down on the ground. And, it works up to 15 feet away. The controller itself is simple – a power switch, the power LED and charge LED, the propeller speed slider and the directional knob. You hold the knob in the direction you want to go, and the horizontal tail prop turns on and off to get the sideways movement desired. The one tail prop prevents the body unit from rotating, and the other contributes to directional movement. The main styrofoam blade gives you lift, and it maintains its height pretty well. The drone is light, at 12 grams, and if it bumps into something, it’ll just bounce away without anything getting damaged, including the styrofoam blade. The kit dimensions are 9.8″ x 7″ x 1″.

Overall, it’s a nice little toy, and is fine for use indoors, but the $39 price tag IS on the high side. Especially when you look at the magazine. This is one of the thinner volumes in a long time. It’s only 36 pages. The first section is a 4-page photo essay with the model/idol talent, Riina flying the drone in a house. This is followed by 6 pages of explanation for how the drone works and how to fly it. There’s 4 pages for building the controller, and 1 page of troubleshooting Q&A. 2 pages of photo essay for the shapes of tree seeds, and 2 pages for an interview with a Japanese drone racer. The editor-suggested mods are to replace the blade with balsa wood, and to put LEDs on the main blade and connect the controller to a PC via an Arduino box for computer-controlled light art. The last 5 pages are an explanation of what drones are, and what uses they’re being put to. There’s no manga this time, no science, and very little theory. There’s also no mention of any future kits.

(Bottom side of the drone.)

I get the feeling that Gakken is having trouble figuring out how to make money on their publications, and they’re cutting corners on projects that appear over-staffed or over-promoted. This is a shame because I like building these kits, and I’d love to see more of them in the electronic music series. Oh well. Anyway, I recommend the Kaeda drone if you can get it in Japan at cover price, without the import mark-up.

(The drone, plugged into the controller to recharge the lithium polymer battery.)

Direct youtube link

Kaeda Drone now out

Gakken finally updated their website to include the regular advertising for the new kit. So, if you want to see what the kit consists of, and get an idea of how hard it is to build it, you can check out the construction sheets. I expect to see this kit arriving in Kyushu either tomorrow, or Friday.