Cube to square answer for Wednesday

Answer for this week:
If the longest line that can fit into a unit square is the diagonal, with a length of sqrt(2), and the largest square that can fit into a unit cube has an area of 9/8 and a side of 3/4 * sqrt(2), then what’s the largest cube that can fit into a tesseract?

First, the largest square that can fit into a cube, AKA Prince Rupert’s Cube, is going to have corners that are exactly 1/4 of the distance from the corner of the cube. For a unit cube, the length of the side of the square is going to be sqrt((3/4)^2 + (3/4)^2) = sqrt(9/16 + 9/16) = 3/4 * sqrt(2). Then the area of that square is (3/4 * sqrt(2))^2 = 9/16 * 2 = 9/8.

The answer for the largest cube that can fit in a tesseract can be found in the Generalizations section for the Prince Rupert’s Cube wiki entry, which was found by Kay R. Devicci, and was also mentioned by Gardner in the addendum to this chapter.


Colossal Gardner, ch. 13

Hypercubes. George Gamow and Martin Gardner both put a lot of effort into describing higher dimensions, since the idea of curved (or warped) space is tied to universe systems of 4 dimensions or more. While String Theory never shows up in Gardner’s book, there is one mention of M-Theory in the addendum for the Hyperspheres chapter, which speculates on 10 or 11 dimensions, most of them wrapped up in a really small space. If M-Theory ever pans out, it could relate gravity to the strong and weak forces, so there’s a reason to consider 4 dimensions and up kind of seriously. On the other hand, Gamow and Einstein argued that our universe is closed in such a way that a spaceship traveling in a straight line out away from our sun would eventually return from the opposite direction. Gamow’s discussions indicate that there might be a left-for-right mirror flip along the way, so that the pilot would be able to put a left-handed glove on his right hand (or that he might be upside-down compared to when he left).

Most of the discussions of hypersolids seem to come down to explaining what a tesseract is, and how it would look to 3D observers. If you’re not familiar with the idea, first take a point on a piece of paper. If you extend the point in one direction, you get a line segment – with 2 points, 1 line, 0 squares, 0 cubes and 0 tesseracts. If you take that line segment and extend it in a direction perpendicular to the segment, you get a square – with 4 points, 4 lines, 1 square, 0 cubes and 0 tesseracts. Extend the square in a direction normal to the plane and you get a cube – with 8 points, 12 lines, 6 squares, 1 cube and 0 tesseracts.

(Rotating tesseract from a 3D perspective, from the wiki article.)

The trick is to then extend the cube in a direction normal to the 3D space, which in effect makes 8 cubes that overlap without touching each other (there’s the original cube, 6 cubes extending from the 6 faces of the original cube, and the final cube you get when you stop pushing in the 4th direction) – with 16 points, 32 lines, 24 squares, 8 cubes and one tesseract.

You can calculate the number of faces, etc., for a hypercube of any dimension “n” by using the simple binomial (2x+1)^n. So, for a 4D cube, write out (2x + 1)(2x + 1)(2x + 1)(2x + 1):
(4x^2 + 2x + 2x + 1)*(4x^2 + 2x + 2x + 1)
(4x^2 + 4x + 1)*(4x^2 + 4x + 1)
16x^4 + 16x^3 + 4x^2 + 16x^3 + 16x^2 + 4x + 4x^2 + 4x + 1
16x^4 + 32x^3 + 24x^2 + 8x + 1
(where the x^4 coefficient gives the number of points, and the x^3 coefficient gives the number of squares, etc.)

To go to a 5D hypercube, just multiply the above equation by (2x + 1) again.

The rest of Gardner’s article is just an examination of what the hypercube would look like to us if we use 3D projections or slices. He goes on to give examples of hypercubes in art, including Dali’s Corpus Hypercubus, and Heinlein’s “-And He Built a Crooked House” short story.

(Dali’s Corpus Hypercubus, from the wiki entry.)

Gardner closes by saying that Heinlein’s idea of the house dropping out of 3D space could have a possible analogue in giant stars that undergo gravitational collapse, according to J. A. Wheeler. The density of a quasar could be great enough to curve space to the point where the mass drops out of space-time “releasing energy as it vanishes.” This might explain the enormous amount of energy emanating from quasi-stellar radio sources.

If the longest line that can fit into a unit square is the diagonal, with a length of sqrt(2), and the largest square that can fit into a unit cube has an area of 9/8 and a side of 3/4 * sqrt(2), then what’s the largest cube that can fit into a tesseract?

(All rights belong to their owners. Images used here for review purposes only.)

Sangaku Flexagon

(Sangaku, from

In the course of writing my chapter synopses of the Martin Gardner Colossal Book of Mathematics, I decided that I wanted to make some flexagons. My net search turned up, which has some very nice free artwork for downloading. What I found most interesting, though, was that one of the patterns was for a sangaku flexagon. Sangaku was the Japanese equivalent of western calculus, discovered about 10 years before calculus appeared in Europe. It was practiced by the upper educated class strictly for its own sake, using wooden Chinese blocks of fixed area to solve advanced math problems. The thing is, the Japanese never found an application for these solutions, instead painting them on wooden planks to be hung up inside of Shinto shrines, dedicated to one of the local gods. I first learned about sangaku while reading volume 38 of the Q.E.D. manga. I’ve been interested in it ever since. Unfortunately, it seems that the sangaku craze never made it down to the southern end of Kyushu, because none of the big shrines down here have ever located sangaku planks in their inventory.

So, this flexagon may be the closest I’ll get to it.

As a quick note, flexagons are folded strips of paper in a hexagonal shape. If you pinch one corner and pull the opposite one, you can turn it inside out to reveal a third face.

4D Answers

How did Slade pull off both tricks?

The braid is something that British Boy Scouts used to make for lanyards, and you just twist one end through the slits.

The looped bands is more of a magic trick. First, start out with a round band and flatten one section. Twist it as shown in the center image, then make the rest of the ring flat. Finally, cut along the length of the now flattened band to make the two loops. The trick is that you show the audience a plain, normal band, then as you pretend to reach into “the fourth dimension,” substitute the gimmick band for the first one.

Colossal Gardner, ch. 12

The Church of the Fourth Dimension is an April Fool’s fiction piece, where Gardner pretends to happen upon a secret cult of 4D worshipers during a visit to London. While most of the article is just an attempt to describe what such a church building would look like, he does mention a couple tricks that the priest, a man named Arthur Slade, supposedly performs to “prove” that he can slip between the 3D and 4D worlds. One is taking a strip of leather with 2 parallel slits in it and braiding it. The other is taking a rubber band and twisting it into two smaller looped bands.

How did Slade pull off both tricks?


On the nature of coincidences

Let’s start with the old chestnut – “If a tree falls in the forest, and there’s no one around to hear it, does it make a sound?” According to the dictionary, “sound” is the vibrations traveling through a medium that are detected by the ear. (I broadened the definition a bit to allow for vibrations traveling through water or an iron bar.) Based on the dictionary definition, the answer is “no.” The tree falling does indeed generate vibrations in the air, but we require a working ear to say that those vibrations are “sound.” (Or, how about changing the question to, “If someone hearing impaired has their hearing aid turned off, is there a sound?” The person is there to see the tree fall, but still doesn’t hear it.) If we change “sound” to “noise,” does this make a difference to our answer? No, because the dictionary definition of a noise is “a sound, especially a loud or unpleasant one.”

However, if we look at that tree, it has obviously fallen over. And, to be precise, if we had taken photos of the trees over an extended period of time, we’d be able to point to the tree in one picture where it is still standing, and another where it is on its side, and say, “yes, it fell over, it didn’t grow on its side like that, and God didn’t make the world from scratch starting with the tree already on its side.” But, unless you’re a lumberjack, or standing in the woods in the middle of a strong storm, it’s very rare for someone to be walking through the woods exactly at the time that a tree within hearing distance decides to fall over. We would say that that’s quite a coincidence; that someone was in the woods to witness the tree falling incident.

But, is it? People walk through wooded areas all the time, and if there are enough trees in the area, then branches and whole trees drop to the ground often enough to be considered a measurable occurrence. And, sometimes these two thing coincide. That is, they happen at roughly the same time and place. Strictly from a statistical standpoint, we can set a probability for someone being at the right place at the right time, through random chance. Under these conditions, is it still a coincidence that we are the ones in the right place to witness the event happening? Not really. We can be amazed by the event; we can be amazed by how unlikely it is that we are the ones to witness the event, and not someone else, and we can comment on how we might have missed the event if we’d been walking by a little earlier or a little later, for whatever reasons. But, deterministically, the event itself was inevitable.

Let’s take this one step further with another chestnut. “How many people do you need in a party so that there’s a better than 50% probability that two of them will have the same birthday?” The answer is 23. Think about that. How many times have you been in a group, or worked in a company with people broken up into groups, where there have been more than 23 in the group? (Things are more interesting if the groups don’t always have the same 23 people all the time.) In how many of those gatherings did everyone announce their birthdays to see if there were any pair-ups? If the answer is “none,” then can we say there was a coincidence?

That is, an outsider tracking everyone’s personal data and movements may know which two people were in the same room at the same time and had the same birth day, but the “coincidence” only comes in when those two people tell each other, or learn about it in some other way. We can then say that “a coincidence” is simply any random event that we have been made aware of.

In a sense, this is the same wording as for “a sound.” If we’re not aware of it, it’s not there. And increasingly, we’re becoming unaware of more and more things happening around us. Keeping our heads down, tuning out everything around us, listening to music players, reading text messages, checking facebook – we’re missing the extraneous random occurrences nearby. When those things do finally impinge themselves on our awareness, either we brush them off as annoyances, or get amazed that they’ve actually happened.

My point is that we live in a random universe, with billions of “entities” (people, animals, objects that are put in motion or become stationary) that are all on their own unique courses. We can invoke chaos theory here, by saying that these entities are following semi-fixed paths around their own “chaotic attractors,” and that statistically, those paths are going to intersect. They are mathematically deterministic. But, unless we “go out of our way” (in an intellectual energy sense, not a physical sense) to pay attention to what’s around us, we’ll miss the things that we can then turn into stories. I.e. – “You know, a funny thing happened to me. It was quite the coincidence…”

All of this is leading up to something of a let-down story. As mentioned in a previous blog entry, I didn’t know anything about Les Paul before reading “In His Own Words.” Asking for a book about him for my birthday was something of a whim, and I wasn’t expecting to get this particular book this time. And, I got 5-6 other books all in the same package, so I could have started reading any of the others first, but I picked up “His Own Words” because I’d thought it might have more to say about electronic sound production. Anyway, one-two days after finishing the book and writing up the review for the blog, the information was still fresh in my mind. I was in the Junkudo bookstore in Kagoshima, up on the 6th floor where they have import books and magazines in English, and the National Geographic collection, 101 Inventions That Shaped the World caught my eye, especially with the picture of the electric guitar on the cover.

So, I thumb through the book, glancing at the various things they cover, and I get to beer. The description of how beer came about and its relation to the production of bread is kind of a sketchy overview, but there’s only 1-2 pages dedicated to each topic, plus photos, so I’m not that surprised that the beer section is so superficial. Then I get to the part on electric guitars, and the main text focuses pretty much only on George Beauchamp’s development of the frying pan. Followed by one sentence saying that other electric guitar manufacturers included “Gibson, Les Paul and Fender.” Otherwise, Les, who made the electric guitar sound popular in his country and jazz recordings, isn’t mentioned at all.

If I hadn’t just finished reading Les Paul’s book, I wouldn’t have looked in the NG book, and I wouldn’t have known that Les Paul was a musician that just liked to tinker with other companies’ guitars to create his own “signature sound,” or that Gibson’s Les Paul line of guitars was just their way of signing on Les as their spokesman. All of which led to my going to Amazon and giving the NG book 3 stars out five for not doing enough fact checking. In other words, National Geographic’s “101 Inventions that Shaped the World” was dead-wrong about Les Paul’s manufacturing guitars, and completing missed the contributions to electric guitars and music recording that he did make.

Was this all just a silly little coincidence, or a random set of circumstances that I just happened to be paying attention to at the time? Whatever. My main point is that the world becomes a lot more interesting when you actively look for these kinds of things, because they are happening all around us all the time. Noise /Off.

Circular answers for this week

1) For the mutually touching circles, if three of the radii have lengths of 1, 2 and 3 units, what’s the radius for circle four for both cases (three circles inside, or outside, of the fourth one)?

x = 6/23 for the small circle, 6 for the large one (although the math comes out as -6).

2) You have 3 perfectly spherical grapefruits resting on a counter, each touching the others. Under them you have a smaller orange also resting on the counter. The grapefruits have a radius of 3 inches. What’s the radius of the orange?

Pretend that the counter represents a fifth sphere of infinite radius, meaning that it has 0 curvature and drops out of the equation. We can now use the same formula as before:

Use the larger of the 2 values from the quadratic equation to get 1 inch.

Colossal Gardner, ch. 11

Spheres and Hyperspheres.
Gardner starts out with a simple definition of a circle – take a ruler with one end fixed in place and put a pen at the other end. Rotate the ruler around the fixed point and you’ll get a circle, defined in Cartesian coordinates as  x^2 + y^2 = r^2, where r is the radius. Let the ruler move in 3 dimensions and you get a sphere, as x^2 + y^2 + z^2 = r^2. Keep adding terms and you can go to as many dimensions as you want. The surface of the object has a dimensionality of n-1. That is, a circle’s surface is a 1D line, and a sphere’s surface is a 2D plane. In the late 19-century, many mathematicians and physicists felt that gravity and electromagnetism in our 3D universe could be transmitted over the surface of a large 4D hypersphere containing the universe.

Einstein himself suggested that within a hypersphere, a rocket moving on a straight line traveling far enough in one direction would end up reaching Earth again from the opposite direction. This concept of an enclosed warped space also shows up in the writings of George Gamow. It ties to the idea that space is warped, rather than flat. Since we’re within space as 3D beings, to us straight lines look straight because we’re tied to the same “surfaces” that the lines are painted on, much like how an ant on an inflated balloon is going to think that the line on the balloon is flat. Looking down on that 3D surface from the point of view of a 4D observer, it’d more obvious that things aren’t flat after all.

Circles and hyperspheres share other properties as well. n-spheres rotate around an n-2 space. Circles rotate around a point, spheres around an axis line, and 4D spheres around a plane. Cross sections are n-1: Cut a circle and you get two points; cut a sphere with a plane and you get a circle; cut a 4D sphere with a 3D plane and you get a sphere. Just like you can turn a thin rubber ring inside out, from 4D space you’d be able to turn a 3D sphere inside out.

(What’s the size of circle #4 in both figures?)

We then move to the maximum number of mutually touching n-spheres. On a plane, you can have no more than 4 circles touching together, either with all 4 being external, or 3 within the larger outer one. The formula is 2(a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2, where a, b, c and d are the bends (curvatures) of each circle, where the curvature is 1/radius. Generalized, the maximum number of mutually touching spheres is n+2, and n times the sum of the squares of the bends equals the square of the sum of the bends.

For packing problems for 3D balls, refer to last week’s blog entry. Generalized answers for N-spheres had not been found at the time of Gardner’s book. He does mention a paradox found by Leo Moser for 9-space, where a 9D sphere in the center of a sphere cluster actually sits outside the cluster while the 2^9 = 512 corners of the enclosing box still have room to hold 512 unit 9-spheres. (Unfortunately, I can’t find a copy of Moser’s paradox online.)

For the above touching circles, if three of the radii have lengths of 1, 2 and 3 units, what’s the radius for circle four for both cases?

You have 3 perfectly spherical grapefruits resting on a counter, each touching the others. Under them you have a smaller orange also resting on the counter. The grapefruits have a radius of 3 inches. What’s the radius of the orange?

Les Paul in His Own Words (and anniversary)

(All rights belong to their owners. Images used here for review purposes only.)

Les Paul in His Own Words, by Les Paul and Michael Cochran
I received quite a few books for my birthday this time, and this is one I’m really excited about. The original hardback came out in 2009, while the centennial paperback edition was released in 2016. To start out with, though, I have to be honest. I’ve only dabbled with acoustic guitar a little, back in the ’80s, and while I have a beginner’s bass right now, I haven’t had many opportunities to practice with it so far (I’m trying to change that and get into the habit of picking it up for at least a few minutes a day, anyway). My real interest has long been with synthesizers, but after reading the book on Robert Moog, I felt that the next big legend to visit in electric music would have to be Les Paul. Otherwise, I knew very little about him what I started out.

Les was a musician first, and a tinkerer second. He was born June 9, 1915, in Waukesha, Wisconsin, as Lester William Polsfuss. His mother tried to teach him piano, but after learning harmonica, he switched to the guitar. He was playing country music by age thirteen, going by the names Red Hot Red and Rhubarb Red, but he also had a strong interest in Jazz. He dropped out of high school, and joined traveling bands that took him to St. Louis, then Chicago, and ultimately he made the move to New York City as part of his main goal of getting good enough to play for Bing Crosby. Eventually, he took on the name Les Paul, and played with the Les Paul Trio (generally with a second guitarist and a bassist, and the line-up would change over time). He moved to California, rented a house that was large enough to hold his own hand-made recording equipment, got drafted and ended up in the Armed Forces Radio Network. He did convince Bing to hire his group to play in the Crosby orchestra, and Bing helped him out a few times, including buying him his own Ampex reel-to-reel tape deck (Bing invested $50,000 in Ampex in part based on Les’ comments about tape being the next new recording format).

Les had married once, when he was still in the Wisconsin area, but his wife couldn’t handle his touring schedule. Between them they had 2 children. Later, he met Iris Colleen Summers, who had been singing as part of the Sunshine Girls. Les convinced her to join his group under the name Mary Ford, and they soon started dating. Les’ first wife agreed to a divorce, although Les and Mary didn’t get married until some time later. Les was working on the concept of multi-track recording with the Ampex, and he and Mary started releasing records as Les Paul and Mary Ford, on the Capitol label. They had several major hits, which attracted the attention of Listerine company, which in turn sponsored the Les Paul and Mary Ford at Home TV show on NBC from 1956-57, then syndicated to 1960. The TV show was unique in that Les decided to sign the contract to air five 5-minute pieces per day, 5 days a week, plus re-editing the show to also run on the radio. The show was recorded in their new home in New Jersey, and new additions were being built on to it all the time to house the TV crews and equipment. Unfortunately, Mary had a miscarriage, and between the subsequent depression and the stress of always being on the go, or surrounded by TV people, on top of suffering from constant stage fright, she divorced Les in 1964. She had drinking problems, got remarried, then died in 1977 in a diabetic coma.

After the divorce, Les kind of went into a tail spin, and his recording career ground to halt. He’d put his body through a lot, from going several days in a row without sleep in order to play radio dates and work on his ideas for an electric guitar, to  accidentally grabbing the power cable of an amateur broadcast transmitter he’d built, while still grounded, and having his muscles ripped apart when they seized up. He’d been in a really bad car accident (Mary had been driving in a snow storm and the car spun off a cliff when she hit a patch of ice) that destroyed his right arm; when it was rebuilt it was stuck in a permanent right angle. Later, he had a heart attack that almost killed him, and as he was getting older, he lost the movement of all parts of both hands, except for his left pinky finger. Through mishaps with friends (both of whom clapped him on the side of his head from behind as a greeting), he ended up getting both eardrums blown. Through all of this, he remained optimistic, and through a huge amount of good luck and the assistance of the many friends he’d made, he still enjoyed himself, and kept playing guitar on Monday nights with a jam group at the Iridium Jazz Club up until his death at age 94 on Aug. 12, 2009.

Notice that Les Paul’s musical career was in country music and Jazz, and had come mostly to an end by 1965, when I was still only 8 years old. To me, his was just a name on the Gibson line of electric guitars, and I still haven’t picked up an electric lead guitar yet. So, after reading the Moog book, I was looking at Les more as an electronics innovator and not as a musician, and I had no emotional baggage with me regarding his musical past. In His Own Words is a pretty unvarnished look at Les’ life, and the primary focus is on his music and the musicians that he played with. On the other hand, he does talk about how his desire for a “Les Paul sound” drove him to create his own electric guitar, his own plate cutting recorder, multi-track recording, and tape-based delay. He doesn’t go into any real technical details for how his gear worked, but he does throw terms like impedance and harmonic feedback around expecting the reader to know what they mean.

And there are guitars. Photo after photo of guitars, promo posters for various musicians and singers, and pictures of himself, his mother, Mary Ford, and his two kids from his first marriage. Plus, he talks about his interactions with Gibson and Fender, and how Gibson came about to produce the Les Paul signature line of guitars.

From a history viewpoint, this book is a nice look at the music scenes from the 1920’s to the 50’s, and introduces Les’ own idols, including Django Reinhardt, and a number of country and jazz greats. It’s an interesting contrast to the Moog book (Analog Days), and is worth keeping in your library. Overall, highly recommended.

To Les Paul: June 9, 1915 – Aug. 12, 2009

Roman Roads of Britain

Following in the footsteps of the Roman subway map, we have England’s metro system. Those Romans. It’s like they’re everywhere.