In my essay Jargons of Software Industry, i mentioned mathematical terms:
polytope, manifold, injection/bijection/surjection, group/ring/field.., homological, projective, pencil, bundle, lattice, affine, topology, isomorphism, isometry, homeomorphism, aleph-0, fractal, supremum/infimum, simplex, matrix, quaternions, derivative/integral …
I'm sure many of you are mystified by these terms. In front of a giant mathematician, i dare not banter these words, but here i think i can give a few explanations to the lay people like yourselves. (thoroughout this essay are littered with relevant URLs for your perusal pleasure.)
In the plane, you have regular _polygons_. That's regularly shaped where each angle are equal, and each side are equal. For example, equilateral triangle, square, regular pentagon, hexagon, heptagon, octagon, nonagon, decagon and so on are all regular polygons. Analogous shapes in 3-dimensions are called regular _polyhedrons_. They are solids where each face is a regular polygon. As it turned out, there can exist only 5 such solids by this requirement, each one has a special name. For example, a octahedron is one of the 5 regular solids. Take two pyramids in Egypt and glue the bottoms together and you get the picture. It's called octahedron because it has 8 faces, each being a equiangular triangle. A cube is also a regular solid, where each face is made of a regular 4-gon the square. Cube is also called hexahedron because of its 6 faces. There's also the tetrahedron (picture it as a 3-sided pyramid, adding the bottom, it has four faces.). Dodecahedron is a regular solid of 12 regular pentagons as faces. Icosahedron is another one made of 20 equilateral triangles.
In summary, the regular solids of 3 dimensions are:
name Faces Edges Vertex ------------------------------------------ tetrahedron 4 6 4 cube 6 12 8 octahedron 8 12 6 dodecahedron 12 30 20 icosahedron 20 30 12
and these are the only ones. It was known (proven) by the Greeks that there cannot be other solids with regular faces. The general name for these is regular polyhedron.
Now, if you have taken multi-variable calculus, you might know that the dimensions can be abstractly extended to more than 3. When, for example, in science fictions they talk about n-dimensions, it is originated and ascertained here in pure mathematics. A 4-dimensional space, from a abstract point of view, is the same multi-variable calculus with 4 instead of 3 variables, all other things remain the same. From that point on, we could derive theorems of all sorts for n-th dimension without knowing what n-th dimension really is or looks or feels like. Nevertheless, the logic is firm, and mathematician's derivative faculties are not delusional.
The details of dimensions higher than 3 takes more space to explain then i have here, but you will proceed to believe me that in higher dimensions, mathematicians have also made systematic imaginations about regular solids. The simplest case many of you must have heard is a hypercube. A hypercube is a 4-dimensional solid that are made of 8 cubes, having 24 squares, 32 edges, and 16 vertexes. A hypercube is also known as Tesseract.
In 4 or more dimensions, the number of regular shapes is finite, just as in 3 dimensions. The general name for regular shapes of any dimension is regular _POLYTOPE_. Thus, a cube is a regular polytope in 3 dimensions.
• Wikipedia.org page on polytope Polytope.
• A Wonderful java applet http://dogfeathers.com/java/hyperstar.html
• George Hart's comprehensive online Polyhedra Encyclopedia http://www.georgehart.com/virtual-polyhedra/vp.html
One can view higher dimensions figures by a process called projection — pretty much like how shadow works. That's also how we can perceive a 3-dimensional object “cube” drawn on a 2-dimensional object “paper”. Another method is by cross-section. Here's a page about projecting or slicing higher dimensional polytopes:
According to the book FLATLAND: A Romance of Many Dimensions, at the last day midnight of every millennium, a being from higher dimension will come down and disclose the gospel of higher dimensions to a chosen one. The unfortunate person (or fortunate depending on your view) will likly go nuts afterwards and be put into asylum for life.
… i digress…
Now, in each dimension there's the minimally configured regular polytope. For example, in 2-D you have equilateral triangle. In 3-D the simplest is the tetrahedron, made of 4 triangles. In 4-D there's pentatope. The generic name for such simplest configured polytope for n-D is called _SIMPLEX_. To me, that term stands for “(math is) simple, but complex!” as in a birch cane licking on some tender snow white ass when explaining the depth of mathematics. Remember that, folks!
Now, with polytope and simplex gone, let me embark on manifold. When i first heard that word, my little mind wondered the depth of mathematics, that i knew mathematicians don't like vacuous jargons as the Unix ＆ Perl programing morons, but they had to invent such mysterious word signify what thoughts that i could not possibly understand without years of study. What thinking! These mathematicians, thinking they do! Unlike philosophers, who think a lot, much oftentimes without a governing body, whereas learned men who think a lot can more or less be philosophers, but whereas in mathematics, lots of thinking won't do. The tremendous thinking and creativity has to fit some kind of nature's rule. The nature being, with a huge sharp ax, lops off thoughts that's won't fit her temperament and exactitude, and without mercy. Therefore, in the fields of philosophy for example, one can have revered crackpots (⁖ Hegel), but not so in math. Now, on to manifold.
Manifold, like polytope, is a generic name for certain concept of arbitrary dimensions. In 2-D, we have curves. For example, take out a piece of paper and scribble on it with a pencil in one stroke. What you have drawn, is a curve, or plane curve. The curves themselves are of 1-dimension, but they sits in a 2-dimensional plane. Now imagine the path of a fly. That would be a curve in space, called space curve. The curve itself is still one dimensional, but sits now in a 3 dimensional space. But now in 3-dimensions, we can have something else beside curves, namely surfaces. The paper you took out earlier, would be a flat surface called a plane, which is a 2-dimensional object, sitting in 3-d space. You can bend the paper or roll up into a cylinder and its still a surface in space. There can be lots of surfaces. A soap bubble, is a specific surface called sphere or hemisphere. A flag is a surface. A open umbrella is a surface. The gist here is that in 3-D space we can have 1-D objects like curves, or 2-D objects like surfaces. As with regular solids of higher dimensions, mathematician's imagination faculties thought about curves and surfaces in higher dimensions. Basically, in n-dimension you can have “curves/surfaces thing” that has dimensions less then n. For example, in 3-D we've seen the 2-D surfaces or the 1-D curves, and in 2-D plane we can have 1-D curves or 0-D dots. Therefore, in 4-dimensional space, we can have not only 0-D dots or 1-D curves or 2-D surfaces, but we can also have “curve/surface-like thing” that is 3-dimensional, sitting in a 4-dimensional space. The general name for these “curve/surface-like things” in arbitrary dimensions is called _MANIFOLD_. Thus, a curve is just a 1-dimensional manifold. A surface is a 2-dimensional manifold. How fantastically fascinating.
With normal math expositors, you can never, ever read such lucid explanation. Instead, you'll bump into esoteric explications which the mathematicians pleased themselves. Only the gruesome Xah is willing and able to bring such beautiful concepts to moron's eyes.
Now, let's talk about injection/bijection/surjection a bit. Except bijection, i don't like these terms. Injection always reminds me of fuel injection, which reminds me of cum. I find these terms a bit pompous, perhaps because i learned alternative terms first.
Injection just means one-to-one mapping. E.g. if we put a spoon in each coffee cup, that's one-to-one mapping because each spoon corresponds to one cup. No spoon, for instance, sits in two cups and no cup holds two spoons. It is “one-to-one”, so to speak. Think of the mappings of penis and p�ssy, then the one-to-one of injection will not be easily forgotten.
A bijection differers slightly in context than injection. Basically, it is a injection with the requirement that for every p�ssy there's is a penis for it. Alternatively, there is no p�ssy that is without a cock. These implies that there are equal number in either party. Bijection is also known as one-to-one AND onto mappings. It is easy to remember because of the symmetry. (think of fairness)
A Surjection is just a onto mapping. I.e. every destination is filled, possibly by more than one entities. For space reasons i'm unwilling to explain further for those who don't have the good concept of function and sets.
As you can see, these names are not as clear as simply saying the mapping is one-to-one and/or onto. Only the bijection is easy to recall.
• Bijection, injection and surjection
The group/ring/field words are fascinating. Normally, math jargons use everyday words. I.e. they try to not invent new words. It is unfortunate that it would take quite some space to explain these well to mere mortals who never heard of them, therefore i won't do it here. But, let me tell you a personal tale about the math term “group”.
Like other math jargons, i was infinitely fascinated by these esoteric terms. I wanted to know what they mean. The mere existence of these terms are sufficient to make me cry, to think that there are god of sorts who have written rules of the universe in stone, and we mortals are left struggling to find them. Just open some graduate math texts — all sort of gibberish talk present themselves. It is these gibberish, that made computers and cellphones and cars and planes and TVs and the survival and sustenance of the massive populace in the last couple of hundred years possible. It is these gibberish that made the lesser beings physicists possible. We could say, that it is these gibberish that made the universe possible. Suppose in a instance all math writings disappear from the face of this earth. Then, you please imagine the consequential catastrophe for a moment. I would suppose, that all today's sciences would immediately cease, even those with experimental methodologies. All engineering, will forthwith stop progress. Moore's law will cease to be true. mm… this would be an interesting topic to put thought on… but i digressed again…
I was talking about the math term groups. A “group” is the name to a abstract concept of: a set and a function that satisfies a few requirements. The study of such are often called Group Theory, and group theory is a branch of modern algebra, where algebra being one of the main trunk of modern mathematics. (the others two are analysis and geometry. (such categorization are very vague.)) The core of group theory, let me tell you, is about the study of symmetry. The previous sentence i just wrote, is one of the fantastic journey and revelation that i personally experienced.
Think about symmetry for a moment. Observe that many animals of earth have bilateral symmetry. In other worlds, their left and right parts are mirror images, or symmetric. There are many other types of symmetry. The patterns on a Persian carpet, is often symmetric. The helical staircase are symmetric. Stairs are also symmetric, in that they repeat on and on. A wire mesh in fences, are also symmetric. Honeycombs are symmetric — look at those regularly arranged hexagons. A equilateral triangle is symmetric too. It has bilateral symmetry along any of the three middle cuts. The peaceful swastika symbol 卍 is also symmetric. The symmetry is what technically called rotational symmetry. It has 4-fold rotational symmetry, in that one can rotate it 1/4th of a circle and the image becomes itself again. The yin-yan tai-chi symbol ☯ also has 2-fold rotational symmetry. Many English alphabets has symmetries. For example, idealized O has all round, rotational symmetric of infinite-fold. X has mirror symmetries of vertical and horizontal axes. H has bilateral, and vertical mirror symmetries. P has none. Here's is a digression i wrote in 1999 about symmetries in letters, modified slightly:
Btw, the name “XAH” is by design. It is not a Chinese transliteration. I choose XAH as my English legal name. I wanted a unique name. Out of myriad letter combinations, XAH came to be more or less because I am fascinated by symmetric alphabets since young. There are still a lot possibilities of names with symmetric letters, but XAH seems to be good and was chosen.
It would be fun to write a program to generate all plausible names that are composed of symmetric letters. Anyone risen for the challenge?
By the way, alphabet may have vertical mirror symmetry (A, M, U…) or horizontal (B, C, E…) or 2-fold rotational symmetry (N, Z, S), or 4-fold rotational symmetry with mirror (X, I), or diagonal symmetry (idealized L, Q), or a combination of the above (H, I, O). We might also consider lower cases. For example C, O, I … has the property that the symmetry is invariant among lower or upper cases. Symmetry are exhibited in whole words in different ways. E.g. AHA, WOW; MOW, pod; DO, COX, BEE, EXCEED; DID, DEED; civic, eve, eye. When applied to sentences in left/right mirror without regards to letter shapes, it's called palindrome. That is, sentences that reads the same forward or backward. E.g. “god”, “A man, a plan, a canal, Panama!”. There are palindromes that are pages long, and I know there are books collecting it as wordy games.
By the by the way, speaking of symmetry… The mathematical study of symmetry is called group theory. Group theory is the abstraction of abstractions. Its beauty surpasses that of visual. Just about few years ago, I was very puzzled by such a idea. How could math, with all the formula and precision, codify such a non-numeric concept as symmetry? What's there to study about symmetry? Isn't it just all ‘looks’? Indeed, knowing what it's about has been a extreme satisfaction for me.
Can you imagine for a second, how could the mundane and everyday concept of symmetry be codified by mathematics to form a complete and far-reaching study called group theory? Groups and group theory is a fundamental concept and extremely important branch of math, and one can get a Ph D specializing in group theory.
As it happens, almost all math that i know i learned on my own. It is around 1997, that i studied the basics of group theory, and got my extreme awe of its beauty. Also, it is a pity of society, that a genius like myself literally took several years to finally understand what group theory is or is about, whereas today i can explain the concept to anyone willing to learn in about a hour, the basics in few weeks. (you now already learned some aspects of what it is about.) Pity the fantastically f���ing bureaucratic schooling system. F��� the system. F��� the stupid teachers. F��� them and f��� their wifes.
(the internet, the info hotbed, to some extend remedies the situation and soothes my anger. I'd like to give a important side-advice here: For the society to progress, there is nothing more important for free-flow of information. I repeat, nothing more important than that. Not prevention of AIDs, not prevention of war, not prevention of deaths or human suffering. Nothing is more important than the massive unfiltered flow of information accessible for anyone, of any kind. A controlled form of such info flow is called education, and a more extreme form of education is called brain-washing. We want free-flow of info. We do not want controlled info. Next time your government speaks of protecting the children or prevention of terrorism with censorship, please furiously kick their face bloody, and vote yourself for governor.)