notes-abstract-categorizations-catChInts

Small integers and some of their properties

The specialest numbers

'Numbers to know', and their properties and where they show up

the special est numbers, the special est shapes

todo: multiplication tables https://en.m.wikipedia.org/wiki/Algebraic_number#Examples https://en.m.wikipedia.org/wiki/Golden_ratio

focus on non negative integers but also a few small positive reals. no negatives, no complex

small algebraic numbers? and all roots for coefficients 1-5 (1-6?); too many; maybe only up to degree 2 (quadratic), mb only monic

https://en.m.wikipedia.org/wiki/Algebraic_number#Numbers_defined_by_radicals https://en.m.wikipedia.org/wiki/Quadratic_irrational https://en.m.wikipedia.org/wiki/Algebraic_number#/media/File%3ALeadingcoeff.png

? https://en.m.wikipedia.org/wiki/Closed-form_expression#Closed-form_number

you can get all the prime factors of the cells in 3dim and 4dim regular polytopes with 2 3 5 -- 7 is not needed. all of 2 3 4 5 6 are factors of the numbers of faces (and hence edges, because there are dual regular polytopes) of various regular 3-polytopes (platonic solids) (and they are the only non-trivial factors of such; if you count trivial factors then you also get 8 12 20 (later: wait what? did i mean 'prime' instead of 'non-trivial'; but then what about 6? did i mean factors with no more than exponent 1 of any prime?); note that the only prime factor here with multiplicity is 2, which goes up to 2^3, reminiscent of the chernoff, also the other very divisible number sequences; however in 4d we have a 600-cell, which has two 5s, but nothing with 2 3).

note that every number below 12 can be reached by some sum of 0 or 1 of each of 1 2 3 5 (another special fact of 12, and another argument for base 12; note that 6 has this property with 1 2 3, and 4 with 1 2, and 2 with 1).

so eliminate 7 from the list of most special numbers. note as special property of 12. note that properties of 3d and 4d convex regular polytopes can be understood without 7, and that these dimensions are special for regular polytopes. note special property of 4,6,8,12,20 wrt regular 3-polytopes, and note d&d dice bc fair dice .

so most special numbers are 1-12, with 1-5 as more special than most (and lots of composite ish special things about 6 as well) (and 1-3 most interesting of these eg cubic, and of course 1-2 most interesting of those eg quadratic https://en.wikipedia.org/wiki/Quadratic and binomial https://en.wikipedia.org/wiki/Binomial_(polynomial)). but MB include 7 as 'first boring number'! also 7 is round of e^2 ( and note 3 is round of e and of pi, and 6 is round of 2pi). 16 (2^4, 4^2) and 20 (4*5 also platonic solid) and 24 (very divisible, also 24-cell, also factorial) and 25 (5^2) and 30 (very divisible (wait is it? it is primorial though); product of first 3 primes 2 3 5) and the primes in there, 13 17 19 what else? gets left out, or put in a section at the end, wherein we skip over boring numbers. note the other maxs (4 6 etc) in very divisible sequences. in misc section also include some more schn/ca numbers eg 360, and powers of primes (8 16 9 27 etc).

4 is the number of sides of a 3-simplex

other popular/fundamental Oeis sequences. also, cite the paper on the oeis frequency distribution to argue that squares and very divisible numbers are important.

note more primes than squares

prime factorization, very divisable number sequences, regular polytopes (most symmetric) (in general, properties of n-space). also lists of squares, primes, powers of small primes, rounded powers of e.

in every dimension, there is a hypercubical regular uniform space filling tiling, but in 2d and 4d, there are a few (2?) others too

another recitation of what is special about every integer through 12:

note: you can see here that the 'claim to fame' of any number after 6 is less solid; 8,9,10,12 all rely on more complex properties that only seem special because we already think that 2, 3, 4, compositeness (which we might identify with 4), and having lots of divisors (which we might identify with 6) are special. 7 and 11 are identified with absences and gaps and dischord (although so is 5). One might argue that 4, 5 and 6 share the same problem, and so the only truly interesting numbers are 0,1,2,3. Or, one might argue that 4 gets in because it is the first number whose specialness relies on the specialness of numbers below it (the first composite), and then 5 gets in as the first 'boring' number. Or one might argue that we should at least go up to 7 because it is the last odd number such that all odds leq it are prime, and because 6 is pretty interesting (the first SHCN and colossally abundant (CAN) after a gap), even if 5 is a little boring; or up until 9 because it demonstrates the first odd number that isn't prime. Personally, i think 0-3,0-4,0-5,0-6,0-7,0-12 have the best arguments. I lean towards 0-12 because i think a good radix should have lots of divisors, so 2, 6 or 12 are the obvious choices; 2 is less than 3, an obviously interesting number, so that's out; 6 is less than 10 so, relative to base 10, base 6 may be seen as 'inefficient' and unlikely to be accepted as a radix in our society. One can also make the argument that, even if only 0 thru 6 are truly interesting, it's a useful exercise to reflect on 7-12 because they are like the 'second coming' of 1 thru 6 (7 is like 1: very odd, centered, reaching; 8 is like 2: very even, filled with 2s; 9 is like 3, the 'second coming' of various numbers based on 2s, also it is filled with 3s; 10 is like 4, notably only for being the first to be made up of others in various ways (note that i think the real second coming of 4 is 16, however; also, 10 has some things in common with 6, as it is 1+2+3+4, so 10 is actually kind of an oddball); 11 is like 5, prime/very odd/reaching and the first such after some kind of gap; 12 is like 6, having many factors (SHCN and colossally abundant), and fittingly, is also = 6*2. By reflecting on 0 thru 12, you get some practice seeing how some of the fundamental themes expressed in 0 thru 7 play out, and you see some phenomena which might serve as counterexamples to hypotheses you might have held after only seeing 0 thru 7 (for example, you see that not all odd numbers are prime). So, as a plan of study, i like 0 thru 12.

Of course it would be nice to go up to 13 (the first prime whose odd-gap between it and the preceeding prime is smaller than the previous gap, also a standard torus can be sliced into 13 pieces with just 3 plane cuts, also a Fibonacci number), 20 (the number of faces in a regular Icosahedron which is the 3-D convex regular polytope with the most faces, and the number of vertices in its dual, a regular Dodecahedron which is a 3-D convex regular polytope, and the number of faces in a 5-simplex, a 5-D convex regular polytope), 24 (a factorial (and so order of the symmetry group S4 (which is also the symmetry group of a regular Tetrahedron, a a 3-D convex regular polytope (Platonic solid))), a multiple of 12, the first number with 4 prime factors that are not all the same), 30 (primorial (the first number with three distinct prime factors), the number of edges in a regular Dodecahedron or its dual, Icosahedron, which are 3-D convex regular polytopes, the number of faces in a 4-D hypercube, which is a 4-D convex regular polytope, and number of edges in its dual, a 16-cell (4-orthoplex); number of cells and vertices in a 24-cell, which is a 4-D convex regular polytope), or 60 (a multiple of 12 which is also a good radix (SHCN, CAN), and is the order of the smallest non-solvable group (A_5) which is also the smallest non-cyclic simple group [7] and which is the group of rotational/orientation-preserving symmetries of Dodecahedrons or Icosahedrons (their full symmetry group, however, is A5 × Z2, of order 120), and number of edges in a 6-orthoplex), or 71, which is the largest prime factor found in the order of any of the sporadic simple groups (sporadic simple groups are 26 or 27 (depending on how you classify things) exceptional groups found in the classification of finite simple groups [8]). But that's way too many to get through. Given the short duration of a human life, learning a little bit about each of the first 12 positive integers is probably all the time that can be spared, except for people who want to specialize in this sort of thing.

(I stopped at 12 instead of 13 because i think 12 would be a nice radix, and also because in some ways primes are the LEAST interesting number, having as they do the simplest possible prime factorization, so the increased prime gap between 7 and 11 followed by the smaller gap between 11 and 13 might be looked at as something interesting happening at 9, followed by ordinary boring stuff at 13, rather than as something interesting happening at 13; but in any case already told you what i consider to be the most interesting things about 13 in this section, so in a way i included 13 after all).

Some properties that we'll cover

Factors

primes

very divisible numbers: SHCN, CA first 15 same todo chart, todo enter these into entries chart below, also intersection of HCN and SA

factorials: 2 6 24 120

squares from 2 to 12: 4 9 16 25 36 49 64 81 100 121 144

cubes from 2 to 7: 8 27 64 125 216 343

n^n from 2 to 4: 4 9 256

primorials from 2 to 210: 2 6 30 210

todo more squares than primes: 1/n vs 1/log(n) [9]

rationals with numerators and denominators of small integers (1 thru 5): 1/7 1/6 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5

small powers of small primes:

2: 2  4   8  16    32    64    128   256
3: 3  9  27  81   243
5: 5 25 125

Multiplication tables: multiplication tables in various bases

Geometric

regular polytopes (most symmetric shapes)

regular tilings

algebraic formulas

misc

"a practical number or panarithmic number[1] is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6"

first few practical numbers starting with 2: 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252

applications

mythical/narrative associations/personifications

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i always thought that everyone associated the number "8" with insanity (maybe not literal mental illness, but the slang meaning of the word, 'craziness, to a degree perhaps unpleasant'), perhaps because of it being 'confusing' for being made up of (prime factor-wise) nothing but a bunch of factors of two (like a hall of mirrors).

but a quick google search suggests that it's just me.

so mb i should note some other personal associations:

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http://math.stackexchange.com/questions/785407/ramanujans-personification-of-small-positive-integers

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0

additive identity (x + 0 = x)

anything multiplied by 0 is 0 (x*0 = 0)

1/7

0.14285714285714285...

1/6

0.16666...

1/5

.2

1/4

.25

1/3

.33333...

2/5

.4

1/2

.5

3/5

.6

Often used as a lower supermajority threshold in parliamentary procedure when something greater than 1/2 but smaller than 2/3 is desired.

2/3

.66666...

Often used as a supermajority threshold in parliamentary procedure.

3/4

.75

4/5

.8

1

multiplicative identity (x*1 = x)

sqrt(2)

1.4142135623730951...

equivalently, 2^(1/2)

(irrational but not trancendental, b/c its a sqrt, right?)

"It was probably the first number known to be irrational." [10]

https://en.m.wikipedia.org/wiki/Square_root_of_2

2

2 is the first number after 1.

2 is the first prime number.

2 is a factorial (2!) 2 is a primorial.

2 is a practical (all powers of 2 are practical).

2 as a radix

2 is thought to be one of the best choices of radix (number base) for digital number representation (base 2 is called 'binary').

2 is of course the smallest, and therefore the simplest, possible radix.

A great advantage of using 2 for number representation is its simple implementation in digital logic, ie when the digits are being represented by signal levels on some medium (eg voltage levels in an electronic circuit). Two signal levels can be chosen, denoting zero and one; levels below the 0-level are still zero and values above the 1-level are still 1; only values which are too far 'in the middle' of the levels for zero and one are indeterminant (if the medium has an indeterminant level when it should be representing 0 or 1, this is an error). The implementation is often done in such a way so that, at the times when the medium is being read, it can be assumed (with high probability) that the level is not indeterminant; therefore determining whether a given medium is in the 0 state or the 1 state is as simple as comparing it to a reference value and asking if it is higher or lower than that reference value.

Indeed, a significant component of the human nervous system is based on binary coding; long-distance transmission of information in our nervous system is primarily accomplished by nerve impulses through axons, which is in the form of 'spikes' (named for their appearance on a graph of voltage vs time), eg at any point in time, a part of an axon can be considered to be either in an 'off' state (no spike) or an 'on' state (spike) (tangentially, the relatively brief duration of spikes (ie unlike voltages in conventional electronic circuits, axons never hold in the 'on' state for an extended period of time; rather, spikes are approximately 'instantaneous') (hence their characteristic shape for which they are named) gives rise to another consideration, that the information transmitted over an axon over time can be approximated by a list of times at which there were spikes, called a 'spike train').

In addition, under the criterion of minimization of r/ln(r) for integers r, 3 is the minimum at 2.731.. but 2 is close, at 2.885.. (see the entry for 3, below, for the motivation for this criterion).

Base 2 multiplication table

The base 2 multiplication table is:

1  10
10 100

e

2.718..

Euler's number (Euler pronounced like 'Oiler'). Transcendental.

"The best-known transcendental numbers are π and e." [11]

todo derivative property

todo defn with limit

3

3-d is needed to form nonplanar graphs without intersections.

3 is the number of spatial dimensions in our world (that we perceive directly, at least).

Any ordered pair (x,y) of two numbers can be categorized into exact one of 3 categories: x < y, x = y, x > y.

Three is the first odd number, which means that it is the first n greater than 1 for which, if there are n entities each voting "yes" or "no" on something, the result will never be a tie.

3 is the first number for which n^2 > 2*n

3 is the second prime number.

3 is the closest integer to the transcendental constant pi (the ratio of a circle's circumference to its diameter), and also the closest integer to the transcendental constant e (Euler's number).

3 is the only number to equal the sum of all the natural numbers beneath it. It is the only number n such that if N = the set of all of the natural numbers less than or equal to n, sum(N) = product(N).

3 is the only prime which is one less than a perfect square.

A lot of the above is from https://en.wikipedia.org/wiki/3_%28number%29

There is a saying "three can keep a secret if two of them are dead". This is probably because, if Person A is certain that only one other person, Person B, knows a secret, and then later finds out that some other Person C knows the secret, they can without further evidence immediately conclude that Person B has told C (assuming that there is no other way to learn the secret except by being told it); by contrast, if three or more people known a secret, and then later one of them finds out that someone else knows the secret, it is uncertain which of the other two confidants. This ease of blame/accountability makes it disproportionately easier for one person to deter another person from telling a secret if the second person thinks that no one else knows.

"Counting to three is common in situations where a group of people wish to perform an action in synchrony: Now, on the count of three, everybody pull! Assuming the counter is proceeding at a uniform rate, the first two counts are necessary to establish the rate, and the count of "three" is predicted based on the timing of the "one" and "two" before it. Three is likely used instead of some other number because it requires the minimal amount counts while setting a rate." -- https://en.wikipedia.org/wiki/3_%28number%29

3 as a radix

3 is thought to be one of the best choices of radix (number base) for digital number representation (base 3 is called 'ternary').

3 is the next smallest, and therefore the next simplest choice of radix after 2.

3 has the advantage of being able to directly represent ternary logic, which comes up a lot; frequently the answer to a question is typically 'yes' or 'no', but can sometimes be 'neither' (neither/maybe/unknown/other/decline to state/does no apply/the question is flawed, etc). In some programming language we have 'nullable types' or 'option types'; in the formal theory of computation we have values called 'bottom' or 'diverging computation' to represent the 'output' of a nonterminating computation. Notice that even in the discussion of the implementation of binary digital logic, we had to mention the possibility of 'indeterminate' signal levels between the 0-level and the 1-level but relatively distant from both of them.

Base 3 can use the 'balanced ternary' representation, where the 3 digits denote -1, 0, 1 (rather than 0, 1, 2), which allows a pleasingly wholistic/symmetric representation of signed integers. It also has some advantages relating to less carrying; in a comment on [12], Chris Lesniewski-Laas says, "Balanced ternary is great for parallel big-number multipliers because you don't need to propagate carry bits arbitrarily far. I used balanced N-ary for my RSA-on-a-GPU implementation.". Knuth calls it the (prettiest? most graceful?) number system in The Art of Computer Programming (perhaps in "seminumerical algorithms" in volume 2) [13]. Some related links: [14] (also at [15]) Ternary computers: the Setun and the Setun-70 in Perspectives on Soviet and Russian Computing: First IFIP WG 9.7 Conference... [16] A Symmetrical Notation for Numbers by Shannon and what seems to be an extremely laborious April Fools Day joke at [17].

There are a few substrates which suggest natural implementation of 3. In optics, dark for 0, and light with two different polarizations for -1 and 1; in electronics, a Josephson junction with off for 0 and clockwise and counterclockwise for -1 and 1 [18]. Interestingly, both of these examples suggest balanced ternary.

Unlike 2, 6, and 12, however, 3 is not a superior highly composite number nor a colossally abundant number nor a practical number.

Base 3 minimizes the r/ln(r) criterion

One proposed criterion for a good radix is the minimization of r/ln(r) (where r, an integer, is the radix being chosen). The motivation is that, in a good radix, it should be efficient to transmit numbers; on consequence is that it shouldn't take too many digits to write numbers. For any given number, the higher the radix, the less digits it takes to write that number; in particular, for a particular number K, the digits required is ceil(ln(K)/ln(r)). If this were the only criterion, then larger radixes would always be better; but consider that the larger the radix, the more information must be transmitted per digit. To transmit the number K, we must transmit ceil(ln(K)/ln(r)) digits, and each of those digits contains ln(r) information (so, if the 'ceil' is neglected, then the choice of radix doesn't affect the information being transmitted in the transmission of K, as expected: ln(r)*ceil(ln(K)/ln(r)).

Now let's say that the expense for us to transmit each digit is proportional to the radix. For example, perhaps for each digit, for each increment of the radix, we must 'set aside' one physical 'spot' that might be used to indicate that this digit is this value. For example, if the radix is 10, then imagine that we represent each digit by having ten holes, and putting a rock in exactly one of the holes to indicate the value of the digit; in this scheme, to represent a 6-digit base-10 number, we'd need to dig 6*10 = 60 holes.

So, in this scheme, the total expense of transmitting a number K is (# of values that might be taken by each digit)*(# of digits) = r*ceil(ln(K)/ln(r)). Neglecting the 'ceil', we see that the effect of r on this cost is approximately to mutliply it by a factor of r/ln(r). That is to say, the loss function we end up with is r/ln(r).

Under the criterion of minimization of r/ln(r) for integers r, 3 is the minimum at 2.731.. . 2 and 4 are the next lowest, at 2.885.. . 5 yields 3.11, and larger radixes continue to yield larger values of r/ln(r) (interestingly, the difference between r/ln(r) as r is increased increases until it reaches it's largest value in the gap 8/log(8) - 7/log(7), and then decreases as r continues to get larger).

Base 3 multiplication table

The base 3 multiplication table is:

1     2   10
2    11   20
10   20  100

pi

3.14159...

The ratio of a circle's circumference to its diameter. Transcendental.

"The best-known transcendental numbers are π and e." [19]

4

4 is the first composite number.

4 is the first square number (2^2), aside from 0 and 1.

4 is the first non-trivial power of the first prime (4 = 2^2).

4 is a power of 2, and a practical (all powers of 2 are practical).

The base 4 multiplication table is:

1     2   3   10
2    10  12   20
3    12  21   30
10   20  30  100

5

5 is the first prime number preceded by a composite.

5 is the first number n for which, if there are n entities each voting "yes" or "no" on something, the result will never be a tie, and furthermore in between 'majority' and 'unanimity' there exists the possibility of 'supermajority'.

5 is the lowest degree of polynomial for which there exists no closed-form formula for finding roots. (todo doublecheck)

6

a primorial.

3! = 6 (6 is 3 factorial, or 6 = factorial(3))

6 is the first composite number with more than one distinct prime factor.

6 is a practical.

"Six is a unitary perfect number, a harmonic divisor number and a superior highly composite number, the last to also be a primorial. The next superior highly composite number is 12. The next primorial is 30."

"All primes above 3 are of the form 6n ± 1 for n ≥ 1."

"Additionally, since the smallest four primes (2, 3, 5, 7) are either divisors or neighbors of 6, senary has simple divisibility tests for many numbers."

"Senary is also the largest number base r that has no totatives other than 1 and r − 1, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table. This property maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its factors do." (totatives are co-primes less than the number)

"Each regular human hand may be said to have six unambiguous positions; a fist, one finger (or thumb) extended, two, three, four and then all five extended.

If the right hand is used to represent a unit, and the left to represent the 'sixes', it becomes possible for one person to represent the values from zero to 55senary (35decimal) with their fingers, rather than the usual ten obtained in standard finger counting...

Which hand is used for the 'sixes' and which the units is down to preference on the part of the counter, however when viewed from the counter's perspective, using the left hand as the most significant digit correlates with the written representation of the same senary number. Flipping the 'sixes' hand around to its backside may help to further disambiguate which hand represents the 'sixes' and which represents the units.

Additionally, this method is the least abstract way to count using two hands that reflects the concept of positional notation, as the movement from one position to the next is done by switching from one hand to another. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. As senary finger counting also deviates only beyond 5, this counting method rivals the simplicity of traditional counting methods, a fact which may have implications for the teaching of positional notion to young students.

In NCAA basketball, the players' uniform numbers are restricted to be senary numbers of at most two digits, so that the referees can signal which player committed an infraction by using this finger-counting system.[1]

More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99, 1,023, or even higher depending on the method (though not necessarily senary in nature). The English monk and historian Bede, in the first chapter of De temporum ratione, (725), titled "Tractatus de computo, vel loquela per gestum digitorum,"[2][3] allowed counting up to 9,999 on two hands. ...

Despite the rarity of cultures that group large quantities by 6, a review of the development of numeral systems suggests a threshold of numerosity at 6 (possibly being conceptualized as "whole", "fist", or "beyond five fingers"[4]), with 1–6 often being pure forms, and numerals thereafter being constructed or borrowed.[5] "

"Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations."

"From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest number of hexagons. This means that honeycombs require less wax to construct and gain lots of strength under compression."

"24-cell: a four-dimensional figure which, like the hexagon, has orthoplex facets, is self-dual and tessellates Euclidean space"

"There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up"."

there are six sides on a cube (2 polarities times 3 dimensions, i guess)

hexagon "Symmetry group Dihedral (D6), order 2×6"

a cube, or hexahedron, is one of the 5 platonic solids

"The cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry)."

" Hexagons are one of only three regular polygons to tessellate the Euclidean plane (along with squares and triangles). The hexagonal tessellation is combinatorially identical to the close packing of circles on a plane. Hexagons are the only regular polygon that can be subdivided into another regular polygon. (As far as I know, hexagons are the only regular polytope of any dimension with this particular property.) In a related fact, hexagons are the unique regular polygon such that the distance between the center and each vertex is equal to the length of each side (sharing this property with the cuboctahedron in 3-space). Hexagons are the first polygons—when ascending by number of sides—that do not form the faces of a regular convex polyhedron in Euclidean space. The three polygons with fewer sides compose the surfaces of the five platonic solids, but no polygon with six or more sides can be employed for this purpose. A consequence of this is that no regular polytope, in any dimension, has hexagonal faces—though many have hexagon-like or hexagonally-symmetric vertices or other elements. Hexagons are third-order permutohedra, meaning each vertex of a hexagon can be described with Cartesian coordinates using one of six permutations of the numbers 1, 2, and 3. Such a hexagon would lie on a plane consisting of all points with coordinates that add up to 6, and would bisect a cube of unit length 2 between coordinates 1 and 3.

Tetractys Another interesting thing about hexagons—and perhaps the most striking fact about them—is that they do, in fact, have six sides. Let us then take a moment to enumerate some interesting facts about the number six:

    Six is a highly composite number, the second-smallest composite number, and the first perfect number. That is, 1*2*3 = 1+2+3 = 6.
    In a related matter, six is the only number that is both the sum and the product of three consecutive natural numbers (1, 2, and 3).
    Six is the smallest composite squarefree integer, and by extension, the first natural number with two distinct prime factors (2 and 3).
    Cubes have six sides, and they are pretty useful too.... The hexagon is of course therefore the highest-sided tessellable regular polygon .... A related property of hexagonal tessellation is that it creates no diagonal edges. "

quotes above probably from one of:

https://en.wikipedia.org/wiki/Senary https://en.wikipedia.org/wiki/6_%28number%29 https://en.wikipedia.org/wiki/Hexagon http://hexnet.org/content/hexagonal-geometry

6 is thought to be one of the best choices of radix (number base) for number representation (base 6 is called 'senary'). 6 it is a practical and it has a large number of divisors for its size (it is superior highly composite number and a colossally abundant number) (it is also a primorial). Compared to 2, however, it is significantly more complex to implement in digital logic (two simple implementation might be: (a) dividing the potential signal levels into a low boundary, a high boundary, and four interior level regions; or (b) representing each senary digit by a hetrogenous mixture of a binary and a trinary digit). 6 also does not score as highly as 3 or 2 under the r/log(r) criterion.

There is a 1/6 exponent in the formula for the F2 Tracy-Widom distribution: https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution

The smallest non-abelian group is of order 6 [20].

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7

r/log(r) has been proposed as criterion for a loss function for choice of radix (see the entry for 3 for a motivation of this). The derivative of r/log(r) is 1/ln(r) - 1/(ln(r))^2 = (ln(r) - 1)/(ln(r))^2. For integer values of r, this derivative reaches a maximum at 7; and the gap 8/ln(8) - 7/ln(7) is the largest such (for integer r).

8

8 is the second number with a composite divisor, and the first number with 3 factors (if you count multiplicity), and the first cubic number greater than 1 (number of the form p^3: 2^3).

8 is the second non-trivial power of the first prime (8 = 2^3).

8 is a power of 2, and a practical (all powers of 2 are practical).

The smallest "extraspecial groups" are of order 8 [21] [22].

"The smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G." is of order 8 [23].

A cube has 8 vertices.

"The spin group Spin(8) is the unique such group that exhibits the phenomenon of triality." -- https://en.wikipedia.org/wiki/8_%28number%29

"The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity." -- https://en.wikipedia.org/wiki/8_%28number%29

Mostly from https://en.wikipedia.org/wiki/8_%28number%29

9

9 is the second non-trivial square number (3^2).

9 is the first non-trivial power of the second prime (9 = 3^2).

10

10 is first composite whose factors skip a prime (2 and 5 are factors, but not the intervening prime 3).

12

12 is a practical.

"The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120."

"The regular dodecahedron can also be represented as a spherical tiling."

"If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges."

"A cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions.[3] In fact, five cubes may overlap and interlock inside the regular dodecahedron to result in the compound of five cubes."

there's a lot of mentions of the golden mean in the properties of the dodecahedron

"The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge"

" The densest three-dimensional lattice sphere packing has each sphere touching 12 others, and this is almost certainly true for any arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions.

Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant Δ(q) whose Fourier coefficients are given by the Ramanujan τ-function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function. This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at −1 i.e. ζ(−1) = −1/12, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons. "

quotes above probably from one of:

https://en.wikipedia.org/wiki/Dodecahedron https://en.wikipedia.org/wiki/Regular_dodecahedron https://en.wikipedia.org/wiki/Honeycomb_%28geometry%29 https://en.wikipedia.org/wiki/12_%28number%29

12 is thought to be one of the best choices of radix (number base) for number representation (base 12 is called 'duodecimal'). 12 it is a practical and it has a large number of divisors for its size (it is superior highly composite number and a colossally abundant number) (unlike 6, it is not a primorial). Compared to 2, however, it is significantly more complex to implement in digital logic (two simple implementation might be: (a) dividing the potential signal levels into a low boundary, a high boundary, and ten interior level regions; or (b) representing each duodecimal digit by a hetrogenous mixture of a two binary digits and a trinary digit (which to me has a pleasing amount of hetrogeneity/asymmetry)). 6 also does not score as highly as 3 or 2 or 6 under the r/log(r) criterion.

Is 12 the largest feasible choice of radix? The next number above 12 which is superior highly composite and/or colossally abundant is 60.

60 is attractive because, coming from a civilization using base 10, it feels like it hurts to lose the prime factor 5 by moving to another base, and 60 is the first SHCN/CAN which is a multiple of 5 (on the other hand, from a digital logic point of view, the prime factor 5 might be seen as a disadvantage; 2 admits very simple digital implementations in many substrates, and there are a few substrates which suggest natural implementation of 3, and even digits in base 6 and 12 can be represented as bundles of sub-digits of 2 and 3; but i suspect it would less convenient to physically represent 5 in digital logic (however a comment by Derek Peschel on http://research.swtch.com/biquinary gives some examples, and there are plenty of hits for Google:quinary+digital+logic)).

Some say that 60 has been used as a radix by humans (sexagesimal) (eg by ancient Sumerians and Babylonians) but as far as i am aware the actual digits were drawn using sub-bases such as 10 sexagesimal, and paid special attention to the sub-bases ("It is worth noting that although the Babylonian 'scientific' system of computation was sexagesimal, they did go to the trouble of writing multiplication tables for the decimal numbers 100 (1,40), 200 (3,20), 400 (6,40), 500 (8,20), 750 (12,30), and 1000 (16,40) as well as the numbers 300 (5), 600 (10) and 900 (15) which occurred naturally. This indicates the strength of a numerical substrate based on 100. " [24] ). As the size of the multiplication table increases with the square of the radix, the base 60 multiplication table would be difficult for humans to memorize; indeed, Babylonians used a system of learning and making partial multiplication tables and then doing some extra computation at the time of multiplication rather than memorizing the entire base-60 table [25].

Therefore, it seems to me that base 60 is impractical (in the ordinary English meaning of the word, not in the mathematical sense of [26]; 60 is in fact a 'practical number'), at least for humans. So, it seems to me that the best feasible choices of radix are 2, 3, 6, 12.

16

8 is a power of 2, and a practical (all powers of 2 are practical).

16 is the third non-trivial square number (4^2).

16 is the only integer n such that there are distinct integers x,y such that:

x^y = y^x

(in particular, with x = 2 and y = 4: 16 = 4*4 = 2^4 = 4^2)

[27]

16 is in the sequence A014221 in which each term is 2 raised to the power of the previous term:

0, 1, 2, 2^(2), 2^(2^2)

This sequence can represent the length in bits of bitfields such that the previous term contains enough bits to pick out or "address" a single location in the bitfield of the current term (zero bits can address 1 location, 1 bit can address 2 locations, 2 bits can address 4 locations, 4 bits can address 16 locations). It can also represent the number of sets of rank less than n [28] (is an offset needed for that?).

16 is also in the sequence where each term is the previous term raised to the power of 2:

2, (2)^2, (2^2)^2

It has 4 identical prime factors (if you count multiplicity), and it is equal to 4*4.

16 is the third non-trivial power of the first prime (16 = 2^4).

misc

todo SCHN, CA, numbers which are both supercomposite (is that what it's called?) and superabundant

todo primes


n-dim convex regular polytopes

In 2-D, we call them 'polygons'. In 3-D, we call them 'polyhedra'. The general, n-dimensional term for this sort of thing is 'polytope'. In 3-D, there are 5 polyhedra which are convex and regular, and we call them "the five Platonic solids".

" In 2 dimensions, the most symmetrical polygons of all are the 'regular polygons'...there is an infinity of regular polygons: one with n sides for each n > 3. (The cases n = 0,1, and 2 are bit degenerate.)

In 3 dimensions, the most symmetrical polyhedra of all are the 'regular polyhedra', also known as the 'Platonic solids'. All the faces of a Platonic solid are regular polygons of the same size, and all the vertices look identical. We also demands that our Platonic solids be convex. There are only five Platonic solids: The tetrahedron, with 4 triangular faces The cube, with 6 square faces: The octahedron, with 8 triangular faces: The dodecahedron, with 12 pentagonal faces. The icosahedron, with 20 triangular faces:

 In 4 dimensions, there are exactly six regular polytopes. 
 The 'hypertetrahedron' - mathematicians call it the '4-simplex' - with 5 tetrahedral faces:  Some people call this a '5-cell', 'pentatope' or 'pentachoron'. 
 The 'hypercube' - science fiction writers call it the 'tesseract' - with 8 cubical faces: 
 The 'hyperoctahedron' - mathematicians call it the '4-dimensional cross-polytope' or '16-cell', with 16 tetrahedral faces:    Some people call this an 'orthoplex', or a 'hexadecachoron'. 
 The 'hyperdodecahedron' - mathematicians call it the '120-cell' - with 120 dodecahedral faces. ...
 The 'hypericosahedron' - mathematicians call it the '600-cell' - with 600 tetrahedral faces:  ...
 Last but not least, the '24-cell', with 24 octahedral faces. This is a denizen of the 4th dimension with no analog in lower dimensions: 

You might things would keep getting more complicated in higher dimensions. But it doesn't! 4-dimensional space is the peak of complexity as far as regular polytopes go. From then on, it gets pretty boring. This is one of many examples of how 4-dimensional geometry and topology are more complicated, in certain ways, than geometry and topology in higher dimensions. And the spacetime we live in just happens to be 4-dimensional. Hmm.

    There is a kind of hypertetrahedron, called the 'n-simplex', having (n+1) faces, all of which are (n-1)-simplices.
    There is a kind of hypercube, called the 'n-cube', having 2n faces, all of which are (n-1)-cubes.
    And there is a kind of hyperoctahedron, called the 'n-dimensional cross-polytope', having 2n faces, all of which are (n-1)-simplices. "

" In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol {3,4,3,3}.

The dual tessellation by regular 16-cell honeycomb has Schläfli symbol {3,3,4,3}. Together with the tesseractic honeycomb (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space. "

" In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets. "

quotes above probably from one of:

http://math.ucr.edu/home/baez/platonic.html https://en.wikipedia.org/wiki/Template:Polytopes https://en.wikipedia.org/wiki/Template:Honeycombs https://en.wikipedia.org/wiki/24-cell_honeycomb https://en.wikipedia.org/wiki/Tesseractic_honeycomb

see also https://en.wikipedia.org/wiki/List_of_regular_polytopes_and_compounds

note that tetrahedrons, hypertetrahedrons, and simplices are analogs of triangles.

note that cubes, hypercubes/tesseracts are analogs of squares.

note that cross-polytopes/hyperoctahedrons are also called 'orthoplex'. Note that simplices are self-dual, and cubes are dual to orthoplex.

(todo; what is the 2-D orthoplex analog, if any?)

In 3-D and 4-D we also have some convex regular polytopes that we don't have in 5-D and above. Namely, in 3-D we have isocohedrons and dodecahedrons, which are dual to each other, and in 4-D, their 4-D analogs. In 4-D, we also have the 24-cell, which has no 3-D analog (but maybe in some way corresponds to a hexagon, in that it tiles? at least, it is listed in the same 'exceptional' column as hexagon in [32] and in [33]).

"In fact, the 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a polygon nor a simplex. Due to this singular property, it does not have a good analogue in 3 dimensions, but in 2 dimensions the hexagon, along with all regular polygons, are self-dual." -- https://en.wikipedia.org/wiki/24-cell

Note: when i say 'dual' in this section, i think that means you replace each face with a vertex and simultaneously replace each vertex with a face; see https://en.wikipedia.org/wiki/Dual_polyhedron https://en.wikipedia.org/wiki/Dual_polygon

Note that all regular polygons are self-dual [34].

"Two dimensions (apeirohedra) Euclidean tilings

There are three regular tessellations of the plane...Square tiling (quadrille), Triangular tiling (deltille), Hexagonal tiling (hextille)

....

Three dimensions (4-apeirotopes) Tessellations of Euclidean 3-space Edge framework of cubic honeycomb, {4,3,4}

There is only one non-degenerate regular tessellation of 3-space (honeycombs),...Cubic honeycomb

...

Four dimensions (5-apeirotopes) Tessellations of Euclidean 4-space

There are three kinds of infinite regular tessellations (honeycombs) that can tessellate Euclidean four-dimensional space: ... Tesseractic honeycomb ...(Self-dual), 16-cell honeycomb (dual to 24-cell), 24-cell honeycomb (dual to 16-cell) ... The hypercubic honeycomb is the only family of regular honeycomb that can tessellate each dimension, five or higher... " -- https://en.wikipedia.org/wiki/List_of_regular_polytopes_and_compounds

The dual of the 2-D hexagonal tiling is the triangular tiling. The 2-D square tiling is self-dual. In general, the hypercube tilings are self-dual, and in 2-D and 4-D the other two tilings are dual to each other. In 5-D and above the hypercubes are the only ones ("The 5-cubic honeycomb...is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space." -- [35]; "The 6-cube honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space." -- [36] "The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space." -- [37] "The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space." [38]); https://en.wikipedia.org/wiki/Template:Honeycombs lists some other honeycombs but i guess they are not regular Euclidean honeycombs. https://en.wikipedia.org/wiki/Hexagonal_tiling , https://en.wikipedia.org/wiki/Square_tiling

Note that the duals of the polytopes and their associated tilings are different; eg the cubic TILINGS are self-dual, but the cubes are dual to the orthoplexes; the simplices are self-dual, but the triangular tiling in 2-D is dual to the hexagonal tiling.

Note that it seems that the tilings may sometimes be identified with infinite polyhedra, also called 'apeirotopes', which i don't understand. A space-filling apeirotope is called a 'honeycomb'.

see also https://en.wikipedia.org/wiki/Regular_4-polytope https://en.wikipedia.org/wiki/Regular_polytope

https://en.wikipedia.org/wiki/List_of_mathematical_shapes#Two-dimensional_regular_polytopes https://en.wikipedia.org/wiki/List_of_mathematical_shapes#Three-dimensional_regular_polytopes https://en.wikipedia.org/wiki/List_of_mathematical_shapes#Tessellations_2 https://en.wikipedia.org/wiki/List_of_mathematical_shapes#Tessellations_of_Euclidean_3-space https://en.wikipedia.org/wiki/List_of_mathematical_shapes#Five-dimensional_regular_polytopes_and_higher https://en.wikipedia.org/wiki/List_of_mathematical_shapes#Tessellations_of_Euclidean_4-space https://en.wikipedia.org/wiki/List_of_mathematical_shapes#Tessellations_of_Euclidean_5-space_and_higher https://en.wikipedia.org/wiki/List_of_mathematical_shapes#Apeirotopes http://quibb.blogspot.com/2010_05_01_archive.html http://members.shaw.ca/quadibloc/math/fdiint.htm

note that 12 = 3*4 and 24 = 4! = 1*2*3*4




(SHCN/CAN means Superior highly composite number/Colossally abundant number; the first 15 numbers in both of these sequences are the same. SHCNs are a subset of highly composite numbers. CANs are a subset of superabundant numbers. See also primorial. )

---

" The first four numbers symbolize the harmony of the spheres and the Cosmos as:

    (1) Unity (Monad)
    (2) Dyad – Power – Limit/Unlimited (peras/apeiron)
    (3) Harmony (Triad)
    (4) Kosmos (Tetrad).[2]

" -- [39]

---

even 17 is so large that it's hard to hold in your mind:

http://mindingourway.com/respect-for-large-numbers/

---

List of groups of orders 1 thru 12

For each integer from 1 thru 12, there is a cyclic group of that order. We won't list those here:

Note: in the above, names which are usually notated as letters with a subscript has just been written as letters followed by a number. The dihedral group and dicyclic groups have been notated with 1/2 and 1/4 their order respectively, but are sometimes notated with their order instead (changing Dih to D; so Dih2, Dih4, Dih5, Dih6 might be called D4, D8, D10, D12 instead, and Dic2, Dic3 might be called Dic8, Dic12), and the dihedral groups are sometimes notated with Dih instead of D.

As for the cyclic groups, some of them are also in other classes:

The cyclic groups not on the previous list are Z5, Z7, Z8, Z9, Z11.

Note that cyclic groups are abelian. Note that finite abelian groups are either cyclic groups, or direct products of cyclic groups [43]. Note that every group of prime order is cyclic. Note that cyclic groups of prime order are always simple [44] and are always elementary abelian [45].

This leaves Z8 and Z9 as the most 'boring' cyclic groups with order between 1 and 12, inclusive (but interesting because of that).

Out of the groups with orders from 1 thru 12, the only non-abelian ones are the following seven:

Notes on the classes mentioned above:

See also https://en.wikipedia.org/wiki/List_of_small_groups and https://groupprops.subwiki.org/wiki/Category:Groups_of_a_particular_order

and maybe link them, eg link to https://groupprops.subwiki.org/wiki/Alternating_group:A5

List of rings of orders 1 thru 12

(random thing i found:

https://groupprops.subwiki.org/wiki/Nilpotent_Lie_rings_of_order_8 (there do not seem to be other pages on that wiki with nilpotent lie rings of other orders; furthermore, i executed a search like this for each order 1 thru 12: https://groupprops.subwiki.org/wiki/Special:Ask/-5B-5BAlgebraic-20structures-20of-20order::8-5D-5D/ and got the groups listed in the previous section, however the only thing that came up besides groups was this page of nilpotent lie rings of order 8). )

An enumeration of the counts of rings from 1 to 12 is (note: Wikipedia claims[47] that this is really the count of 'rngs', rings without multiplicative identity; i don't know enough to know if that's right or not):

1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, 390, 2, 22, 2, 22, 4, 4, 2, 104, 11, 4, 59, 22, 2, 8, 2

[48] [49] [50]

Note that the this sequence achieves new maximum values at powers of 2 (note that the analogous sequence for groups, [51], also almost has that behavior, except that no new max is achieved at 2 and new maxs are achieved at 24 and 48).

Anyhow, there's no way that i'm going to list all of the 11 rings of order 4 or the 52 rings of order 8, unless they have some repetitive structure, which i can't quickly find, so i guess this is where this section ends for now.

todo

https://www.abstract-polytopes.com/atlas/1/

https://www.abstract-polytopes.com/atlas/2/

... etc

https://www.abstract-polytopes.com/atlas/11/

https://www.abstract-polytopes.com/atlas/12/

" S0 and S1 The symmetric groups on the empty set and the singleton set are trivial, which corresponds to 0! = 1! = 1. In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S0, its only member is the empty function.

S2 This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group and is thus abelian. In Galois theory, this corresponds to the fact that the quadratic formula gives a direct solution to the general quadratic polynomial after extracting only a single root. In invariant theory, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting fs(x, y) = f(x, y) + f(y, x), and fa(x, y) = f(x, y) − f(y, x), one gets that 2⋅f = fs + fa. This process is known as symmetrization.

S3 S3 is the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from S3 to S2 corresponds to the resolving quadratic for a cubic polynomial, as discovered by Gerolamo Cardano, while the A3 kernel corresponds to the use of the discrete Fourier transform of order 3 in the solution, in the form of Lagrange resolvents.[citation needed]

S4 The group S4 is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, 9, 8 and 6 permutations, of the cube.[5] Beyond the group A4, S4 has a Klein four-group V as a proper normal subgroup, namely the even transpositions {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, with quotient S3. In Galois theory, this map corresponds to the resolving cubic to a quartic polynomial, which allows the quartic to be solved by radicals, as established by Lodovico Ferrari. The Klein group can be understood in terms of the Lagrange resolvents of the quartic. The map from S4 to S3 also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree n of dimension below n − 1, which only occurs for n = 4. "

Links

todo: read

http://math.fau.edu/richman/Interesting/WebSite/notes.pdf