#497 Four touching spheres form a tetrahedron – Thank you for the inspiration, Nanodots! – A new minimal geometric composition each day

"The beauty in mathematics is unappreciated by many but matched by very little."

All sorts of things related to mathematics.

About

If you’ve read my posts on the integers and the rationals you’ll see that the naturals aren’t really a subset of the integers, which are not really a subset of the rationals. We can

identifycertain natural numbers with certain integers, sure, but it’s definitely not true that natural 0, {}, equals integer 0, {{{0}}, {{1}}, {{2}}, {{3}}, …}.Unless…

The integersWe could create a new set. This will be the actual set of the integers,

Z. This new set will include both the original setNof the natural numbers and the set of the equivalence classes of the integers - let’s call itB. ThenZ=N∪B.In

Nwe have an equality = which is an equivalence relation between two natural numbers and goes something like this:∀a, b (a = b ↔ [∀c (c ∈ a ↔ c ∈ b)])

Two natural numbers

aandbare equal if and only if all members ofaare also members ofband vice-versa. That’s just the Axiom of Extensionality, really, and it’s valid for any sets. Since the construction of all the naturals is constant and so is that of the integers, that equality is good enough for them. But not for us. So now we’ll define a new equivalence relation inZ:[=](I put it in italics to highlight that this is a special relation whose strict symbol is [=]). It goes like this:∀a, b (a

[=]b ↔ (a, b ∈Z) ∧ [(a = b) ∨ (a ∈N∧ b ∈B∧ ∀c ∈N[(c + a, c) ∈ b]) ∨ (b ∈N∧ a ∈B∧ ∀c ∈N[(c + b, c) ∈ a])])Under this equivalence relation, a natural number

xequals an Integeryif and only if all members of that integer are ordered pairs of the form (k + x, k); a natural number[=]s another if they’re equal; same goes for two integers.So, you see, with

thisset andthisequivalence relation we can properly say thatNis a proper subset ofZ!Now we need to also define our new operations. The easiest way to define them is to make all results be integers instead of natural numbers - that way we won’t get any confusions!

Addition∀a, b, c ∈

Z(a[+]b[=]c ↔ ∀x, y ∈B(x[=]a ∧ y[=]b ∧ x + y = c))In this definition,

[+]is the new addition we’re defining onZ,[=]is the equivalence relation we explained above, + is integer addition onBand = is integer equality onB.

Subtraction∀a, b, c ∈

Z(a[-]b[=]c ↔ ∀x, y ∈B(x[=]a ∧ y[=]b ∧ x - y = c))Once again we’re defining

[-]using integer - and = onB(remember that integer subtraction is defined as addition to the inverse of the second term.)

Multiplication∀a, b, c ∈

Z(a[*]b[=]c ↔ ∀x, y ∈B(x[=]a ∧ y[=]b ∧ x * y = c))Once again, pretty easy, with integer multiplication.

DivisionAaahh… now things get complicated. However, we can just use this and replace the -s with

[-]s and the +s with[+]s and the =s with[=]s et voilá.

Inequality∀a, b ∈

Z(a[≤]b ↔ ∀x, y ∈B(x[=]a ∧ y[=]b ∧ x ≤ y))In this case, ≤ is the inequality defined in

Band [≤] is the one we want to define inZ. We can define the others based on this one. For everyaandbinZ:

- a
[<]b ↔ a[≤]b ∧ ¬(a[=]b)- a
[≥]b ↔ b[≤]a- a
[>]b ↔ b[<]aAnd we’re done with the inequalities.

Set membershipWe can even invent a new kind of set membership,

[∈]! It goes like this:∀s (a

[∈]s ↔ ∃b ∈ s (a[=]b))That is, if we have any integer or natural number

bthat’s a member of some setsthen all numbers that are equivalent tobunder[=]possess the[∈]relation withs.So whenever we talk about the set

Zof integers, that’s the one we’re talking about, and the operations we just defined are the canonical operations we’d expect. So we can indeed add an integer and a natural number and we’ll get an integer. And we can say that the set of natural numbers is a proper subset of the integers. We really have managed to make this set behave exactly as we were taught the set of integers should behave.

The rationalsNow we do the same thing for rationals. Of course things get more complicated, but not too much so. Let’s call the set of the equivalence classes of the rationals… I dunno,

Mperhaps. And the set we’re trying to build isQ=B∪M. Anyway, whenever we do not mark the operations below, we mean the operations on the rationals.

EqualityRational equality is

{=}and it goes like this:∀a, b (a

{=}b ↔ (a, b ∈Q) ∧ [(a = b) ∨ (a[=]b) ∨ ∃c, x ∈B(((c[=]a) ∧ (b ∈M) ∧ ((c*x, x) ∈ b)) ∨ ((c[=]b) ∧ (a ∈M) ∧ ((c*x, x) ∈ a)))])This complicated definition will say that a

{=}b if at least one of the following is true:

- a = b under the general definition of equality in set theory;
- a
[=]b, that is, they’re both integers and are equal;- there is some pure integer
csuch that either a[=]c,bis a rational and is the representation ofaor b[=]c,ais a rational and is the representation ofb.These three cases cover everything: if

aandbare both naturals, integers or rationals, the equality holds; ifaandbare members ofZthat would be equivalent under[=], the equality holds; if eitheraorbis a member ofZand eitherborais a member ofM(the set of the equivalence classes of rationals) then we find a membercofB(which is a proper integer) that’s equivalent to our member ofZand check whether the other one is also the rational representation of the former.

Addition∀a, b, c ∈

Q(a{+}b{=}c ↔ ∀x, y ∈M(x{=}a ∧ y{=}b ∧ x + y = c))In this case,

{+}is the new operation of addition we want to define and + is the operation as it is on the rationals.

Subtraction∀a, b, c ∈

Q(a{-}b{=}c ↔ ∀x, y ∈M(x{=}a ∧ y{=}b ∧ x - y = c))

Multiplication∀a, b, c ∈

Q(a{*}b{=}c ↔ ∀x, y ∈M(x{=}a ∧ y{=}b ∧ x * y = c))

DivisionAnd now we got to the meat of rationals. Since we can this time get the results of all divisions without bothering with silly remainders and whatnot, we can have a proper definition of rational division:

∀a, b, c ∈

Q(a{/}b{=}c ↔ ∀x, y ∈M(x{=}a ∧ y{=}b ∧ x / y = c))We can divide any rationals we want, and now we found an operation that works even when those rationals are also naturals or integers.

Inequality∀a, b ∈

Q(a{≤}b ↔ ∀x, y ∈M(x{=}a ∧ y{=}b ∧ x ≤ y))But wait! We haven’t defined the ≤ operation on the equivalence classes of the rationals yet! How does it work?

It’s a bit more complicated, too. It goes like this:

∀a, b ∈

M(a ≤ b ↔ ∀w, x, y, z ∈B(x > 0 ∧ z > 0 ∧ [(w, x)] ∈ a ∧ [(y, z)] ∈ b → wz ≤ xy))That is, we find a representation of the rationals

aandbas their equivalence class such that the second term of the ordered pair is positive and the inequality on the integers wz ≤ xy follows.With this we can define <, ≥, >,

{<},{≥}and{>}in the usual way.You should pay attention to what kinds of mathematical objects we’re dealing with. Since

w,x,yandzare proper integers, their comparison is the integer one; sinceaandbare proper rationals, their comparison is the one we want to define.

Set membership∀s (a

{∈}s ↔ ∃b ∈ s (a{=}b))It’s the same thing as before: a new kind of set membership that automatically “includes” in all sets that have some rational

qall sets equivalent to it under{=}.

So what?Now that we’ve built those sets, all the properties we’d expect them to possess are there. With this formalisation we don’t need to worry about the set of natural numbers

not really beinga subset of the integers and so on.When we talk about equality, inequality or the operations on those sets, we’ll be talking about the ones defined above. If I say n ≤ z where

nis a natural number andzis an integer, I’ll just be talking about the relation[≤]which is perfectly analysable and perfectly convenient.There are still caveats. There will always be caveats. One example is that whenever we make a set with rationals, for example, we’ll need to get not only the pure rationals but also the integers and naturals that are equivalent to them under

{=}. The relations[∈]and{∈}aren’t really the same thing as ∈, but wecanuse them in a mostly intuitive way and you can also easily see that a ∈ b → a[∈]b and also that a ∈ b → a{∈}b (because both[=]and{=}are equivalence relations) so if the set we’re building includes all “versions” of the same number under[=]or{=}then we’re good.

Posted 11 months ago

#mathematics #mathema #integers #rationals #axiom of extensionality #operations #sets

#mathematics #mathema #integers #rationals #axiom of extensionality #operations #sets

22 notes

(Source: imathematicus)

Posted 11 months ago

#mathematics #mathema #diagram #topological spaces #metric spaces #normed vector spaces #inner product spaces #topology

#mathematics #mathema #diagram #topological spaces #metric spaces #normed vector spaces #inner product spaces #topology

100 notes

Mandelbrot island - Mandelbrot set rendered as an island with Terragen, a fractal-based landscape generator.

Fathom the Universe

Source:http://commons.wikimedia.org/wiki/File:Mandelbrot_island.jpg

(via visualizingmath)

Posted 11 months ago

#mathematics #mathema #fractal #Mandelbrot island #Mandelbrot set #Terragen #interesting

#mathematics #mathema #fractal #Mandelbrot island #Mandelbrot set #Terragen #interesting

268 notes

3D Laser Cut Paper - Geometric Artby Eric StandleyEric Standley is an artist and educator currently living and working in Virginia. In his incredible series of 3D laser cut paper art, Standley’s work is found at the intersection of art, technology, history and

mathematics.His vector drawings were initially inspired by the geometry in Gothic and Islamic architectural ornamentation. The pieces are painstakingly assembled from laser-cut paper, layered to create elaborate 3-D works of art. Often these works are created using well over 100 layers of paper and can take months of planning and drawing. The result is so intricately detailed that the pieces must be viewed from multiple perspectives to be fully appreciated.

Standley uses an array of colors woven together with mathematical precision to create his art, combining 12th century architectural aesthetics with contemporary technology. In designing his pieces, Standley envisions three to seven layers of paper at one time, picturing how they will build upon one another. Source.

Posted 11 months ago

#mathematics #mathema #art #geometric art #3d laser cup paper #Eric Standley #these are gorgeous #paper craft

#mathematics #mathema #art #geometric art #3d laser cup paper #Eric Standley #these are gorgeous #paper craft

1,484 notes

The Prime Pages (prime number research, records and resources)

Access to hundreds of pages of award-winning information on prime numbers—with links to thousands of pages elsewhere. We host the searchable database of the 5000 largest known primes (updated weekly). We also have a glossary, top 10 records lists, prime-music, programs, free downloads, and much more!A very,

veryinteresting website dedicated to primes!

Posted 11 months ago

#mathematics #mathema #resource #prime numbers #this is an excellent resource #for anything related to primes

#mathematics #mathema #resource #prime numbers #this is an excellent resource #for anything related to primes

16 notes

(Source: facuontivero, via mathmajik)

Posted 11 months ago

#mathematics #mathema #geometry #hexagon #perspectives #universal language of mathematics

#mathematics #mathema #geometry #hexagon #perspectives #universal language of mathematics

54 notes