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]]>Some of the more egregious irregularities in math notation include
An irregularity in mathematics that hurts young students is number names in English. The numbers “one” through “ten” are decent, but then things get strange. “Eleven” and “twelve” are only distantly related to “one” and “two”, “thirteen” should really be “ten three”, and “twenty” should really be “two ten”. In Chinese (and Korean and Japanese) numbers are named in an entirely rational systematic way — the equivalent of counting “one”, “two”, “three”…“ten”, “ten one”, “ten two”…“ten nine”, “two ten”, “two ten one”, “two ten two” and so on. The result is that students taught in Chinese master base 10 place value concepts faster and more reliably than students taught in English.
It is impractical of course to change the names of numbers in English, but teachers could use the rational number names as an alternate scaffolding notation during teaching. Elementary teachers commonly use physical base 10 blocks to teach place value — physical “manipulatives” like these can be considered an alternate notation.
The most developed alternate notations for math can be found in programming languages. APL, invented by Ken Iverson in 1960, started not as a programming language, but as a notation for working with arrays. APL does two key things: all notations follow the same syntax (infix operators that can be unary or binary), and all operations can apply to arrays of numbers as well as to individual numbers.
The Wolfram Language — the core of Mathematica — also qualifies as a rethinking of math notation that irons out all syntactic irregularities. The Wolfram Language goes far beyond encompassing just matrices — all mathematical objects are swept into its domain of discourse, including proofs, derivation sequences, and much more.
But the biggest problem with mathematics notation is not syntactic irregularities, but rather all the things that mathematics does not even attempt to notate. For instance, how do you ask “what is 3+4” in purely mathematical notation? You can’t. You must use a mixture of mathematical notation and English. Schoolchildren come to read the incomplete phrase “3+4=” as asking for an answer, with the equals sign meaning “produces the answer”. No wonder students are confused by mathematical notation.
A similar dilemma faces students who want to express the idea of a function that maps x to x+3. You can write “f(x)=x+3”. But is that statement an equation declaring that two things are equal, an expression that evaluates to true or false, or the definition of a function called “f”? Computer languages explicitly distinguish between these ideas, whereas mathematical notation does not.
I consider it strange that mathematics, supposedly the pinnacle of rational thought, is conducted in an unruly mixture of English, symbols and references to other papers. And that most mathematicians themselves do not realize that their house needs to be cleaned makes the problem much worse. There are a few mathematicians who seek to formalize everything and do so, but their notation is even less readable than APL. To my mind mathematicians have a lot to learn from computer science — cleaning house would not just make math easier to learn, it would make many deep truths easier to perceive. And to be clear the reverse is true too…computer scientists have much to learn from mathematicians.
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]]>Block by Block Jr. is a 3d construction puzzle based on the classic Soma Cube puzzle. Like the Soma Cube, Block by Block Jr is a great way to learn to visualize shapes in 3 dimensions — an important skill in science, math and engineering. For instance chemists visualize molecules, mathematicians visualize data, and architects visualize buildings and other structures. Solving Block by Block Jr. puzzles also develops problem solving and teamwork skills.
Every year my daughter’s elementary school stages a day of hands-on science exhibits called Discovery Day. For this year’s event I decided to build a giant 3d puzzle, inspired by giant Soma cubes other people have built out of cardboard boxes. However, when I tested the Soma cube on kids, I found it was much too hard. Adults too had trouble with it.
So I developed a simpler version of the Soma Cube, with fewer, smaller pieces, which I call Block by Block Jr. The original Soma cube has seven blocks that form a 3x3x3 cube. Each block is made of 4 cubes, like a Tetris block, except for one block that is made of 3 cubes. Altogether the blocks contain 27 cubes, and can be assembled to make a 3x3x3 cube.
Block by Block Jr has six blocks: three blocks made of 4 cubes each, and three more blocks made of 1, 2, and 3 cubes. Altogether the blocks contain 18 cubes, and can be assembled to make a 3x3x2 rectangular solid. Block by Block Jr puzzles are definitely easier than Block by Block puzzles, but the hardest ones are still quite challenging. Here are the six Block by Block Jr pieces.
For Discovery day I built three versions of the puzzle: a giant version featuring built out of 10″ cubical cardboard boxes stuck together with duct tape,
a half-sized version made of 5″ cardboard cubes,
and a small version made of 1″ wooden cubes
In all three versions of the puzzle I stuck colored dots onto the blocks to make it easier to identify which block is which.
Here are instructions for making your own set, including where to order cardboard boxes and wooden cubes.
And here are challenge sheets containing puzzles that require 2 to 6 blocks. Caution: there are answers for only the first 5 (of 10) challenge sheets.
You are welcome to make copies of this puzzle for personal use or for gifts, but please do not sell them…I’m working on turning this puzzle into a product.
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]]>SYMMETRY. Reflection about a vertical axis. INSPIRATION. Commissioned by high school mathematics teacher David Masunaga for a talk about origami, and published in Peter Engel’s book Origami from Angelfish to Zen. STORY. This design was created at the request of David Masunaga, who used it in a talk about origami to illustrate the concept of grafting one construction onto part […]
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]]>SYMMETRY. Reflection about a vertical axis.
INSPIRATION. Commissioned by high school mathematics teacher David Masunaga for a talk about origami, and published in Peter Engel’s book Origami from Angelfish to Zen.
STORY. This design was created at the request of David Masunaga, who used it in a talk about origami to illustrate the concept of grafting one construction onto part of another construction. See the previous inversion of the week for the complete story.
The four letters RIGA in the middle of ORIGAMI form the two Chinese characters which are called ORI and GAMI in Japanese. ORI means “to fold,” and is composed of the characters for hand and ax. GAMI means “paper,” and is composed the characters for silk and family.
For comparison, here is the original calligraphic version of origami on which I based my design, written by calligrapher and seal carver Xu Yunshu.
The G and A work rather easily. The biggest liberty was turning the substantial right angled roof on the ax character into the more liquid melting dot on the I. Notice that I repeated the same sort of droopy dot on the other I.
The letters O, M and I are written in the same visual style. I could have written them in purely Roman letters, as I did with Elise Diamond. M and I don’t particularly look like Chinese characters, but O looks strongly like the the word “ko” (“mouth” or “entrance”) in Japanese, so maybe this design looks like someone is eating a piece of origami, or an entry sign at an origami exhibition.
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Superteacher 1998 SYMMETRY. Reflection about a vertical axis. Looks the same in a mirror. |
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]]>Montgomery 1999
SYMMETRY. Rotation by 180 degrees. |
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]]>Meir Yedid 1998
SYMMETRY. Rotation by 180 degrees. |
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]]>J. S. Bach 1981
SYMMETRY. Reflection about a vertical axis. Looks the same in a mirror.
INSPIRATION. Appears in my book Inversions as part of a trio of inversions in tribute to the book Gödel Escher Bach.
STORY. As a pianist, I’ve always been drawn to Bach’s music. I am particularly fond of the canons and fugues in the Well-Tempered Clavier, Musical Offering and Art of Fugue. Canons are similar to inversions — the goal in both cases is to compose an aesthetically pleasing result by following a mathematically precise rule.
I have composed a number of canons over the years. Here is a Canon by Augmentation I composed on the theme of the Musical Offering. There are two voices, which start an octave apart. Both voices play the same notes, but the higher voice plays twice as fast as the lower voice. Notice that the higher voice completes two repetitions in the time it takes the lower voice to complete one. The exact symmetry is broken only on the last note. I wrote this canon as a gift to Douglas Hofstadter when I was helping him teach a course based on the then forthcoming book Gödel Escher Bach.
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]]>Synergy 1981
SYMMETRY. Tessellation with two 90° centers of rotation.
INSPIRATION. Inspired by the work of R. Buckminster Fuller,
STORY. R. Buckminster Fuller, best known as the inventor of the geodesic dome, was an advocate of doing more with less. His watch word was “synergy” — the behavior of a whole not predicted by the behavior of its parts. I am particularly fond of his interest in tensegrity figures (available in toy form as Tensegritoy), gravity defying constructions of sticks and strings which were originally developed by sculptor Kenneth Snelson. You can find out more about Fuller’s work through the Buckminster Fuller Institute.
This design practices synergy in two ways. First, the word crosses itself four times at two different types of junctions: S becomes Y and E becomes R. Second, letters are joined in pairs, reducing the number of modules to just three. I was pleased to find that this pattern fills a grid without leaving any gaps.
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]]>Happy New Year 2001
SYMMETRY. Rotation by 90° turns the first letters of the family member names into HAPPY NEW YEAR.
INSPIRATION. Created for my parents as an annual greeting card.
STORY. Every year since I was in high school I have produced a greeting card for my parents. This card features the names of my parents (Lester and Pearl), my siblings and myself (Scott, Grant, Gail), our respective spouses (Amy, Chilju, Vaughn), and our respective children (Gabriel, Michael & Eliott, Kyra & Liana). Eliott usually goes by his Korean name Han Sol.
I’ve kept family members grouped together, with names listed from oldest to youngest within each family. The only exception is Kyra, who is older than Liana; I couldn’t find a way to list Kyra first. Note that families align neatly with the boundaries of the words HAPPY, NEW and YEAR. Also note that the letters in each name are drawn to match the style of the initial capital.
I produced an earlier version of this design in 1995, in honor of the births of Grant and Chilju’s son Eliott, and Gail and Vaughn’s daughter Kyra. Two years ago Gail had a second daughter Liana, and I had a son Gabriel, so I reworked the design to include two more names. The trickiest part is the overlapping G and L, which make EA.
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Termes
SYMMETRY. Drawn with a single line on the surface of a cube. The line begins and ends at with the crossbars of the two E’s. Can you trace the entire line with your eye?
INSPIRATION. For Dick Termes, artist who paints on spheres.
STORY. I first met Dick Termes in 1992 at the Math Art conference. I had seen his work in books, but flat pictures hardly do justice to his work. You see, Termes paints images of landscapes and environments on the surface of a sphere, not on flat canvas. Instead of a narrow window onto a larger scene, you get the full panorama. And not just a 360° panorama like the walls of a room. You get absolutely everything in all directions, including sky above and earth below.
I find his borderless paintings wonderfully freeing. Instead of being locked into a single frame of reference, you are invited to look around. You can’t see the whole painting at once, so the act of seeing retains a sense of mystery and discovery. The seamless sphere restores a sense of wholeness lacking in a chopped off rectangular frame.
To produce his paintings Termes invented his own system of what he calls six-point perspective. He paints on tough light plastic spheres originally manufactured for light fixtures. After he roughs up the surface with sand paper he applies gesso and acrylic paint. The final spheres are hung from the ceiling on a motorized rotating mount. He has created over 140 Termespheres since 1969.
For pictures and stories about Termes’ work, visit his web site. He has many items for sale. The spheres themselves are expensive, but he also offers inexpensive paper models that fold into Platonic solids, as well as a book on his perspective system and a video.
When I met Dick Termes I knew I had to do something with his name. Naturally I wanted to draw his name on a sphere. His name has six letters, so I reshaped the sphere into a six-faced cube. I placed the letters in a zig-zag pattern so the whole name can be read by turning the cube around a single axis.
Doodling with his name I discovered that the letters connected well one to the next, so I searched for a way to draw the entire name with a single line. I found working with the entire surface of a cube a wonderfully three-dimensional experience. I was pleased to find that the two E’s provided natural beginning and ending points for the line. Only the R face has obviously superfluous lines. I drew the cube in Illustrator, popped it into three dimensions with Adobe Dimensions, and animated it with Flash.
If you would like to make your own “Termescube”, here is a model you can print, cut out, and fold together.
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