Angle
&
Symmetry

You know, angle itself is not exactly a static concept,

being a measure of the degree of rotational travel

between two line positions.

When two lines meet at a point,

you distinguish between the acute and reflex angles

by imagining the amount of travel between one line position and the other.

And that ability to imagine motion

and freeze it in a lower dimension

is something that humans use all the time,

not just for mathematics.

But when looking for beauty in collections of angles,

it is obviously not just found in Islamic tilework.

There can be beauty in game worlds,

the 2d screen allowing us to imagine moving through a myriad polygons,

finding rhythm in motion and balance,

and sometimes that most perfect form of balance,

like mirror calligraphies,

symmetry.

Humans may have to begin with an understanding of symmetries of simple 2d shapes and 3d solids,

but maths does what it does so well,

and ignores the things we can see and touch,

and sets off with its abstraction to imagine how this all works in other dimensions,

expanding simple symmetry

into its own land of exploration,

a land the exploration of which led to the discovery

of something very, very, very strange

The Monster.

Now a 2d square can be imagined into a 3d cube,

extruding it into our familiar 3d,

but it's harder to imagine things

that are in dimensions beyond our experience.

So a 4d hypercube

is a lot easier to mathematically describe

than it is to picture.

But once having abandoned images to follow the numbers,

a vast and bizarre universe is opened up.

Just as the number of possible symmetries

expands as a square grows into a cube,

with its 48 different symmetries,

symmetry group 48

other shapes can also have the same number of symmetries.

The octahedron is also symmetry group 48,

and you can see how the two shapes are inside each other,

when you connect the central points of adjacent faces.

But even for a square, its number of symmetries

can be broken down into subgroups of different kinds,

rotational and reflective symmetry,

repeat the rotaton four times

and you are back where you started,

but reflection only needs repeating twice.

So it is that almost all symmetry groups

can be built up from smaller groups that are indivisible,

much like numbers can be built up from primes.

The search for these simple groups,

atoms of symmetry,

indivisible symmetry groups in other dimensions,

has occupied the attentions of a great many mathematicians.

But The Monster stayed well hidden,

far beyond the imagination of mathematicians for a long time.

The history of mathematical understanding

as it moves across the centuries into new and unknown places

is really just the story of those mathematicians that imagined it,

and did the calculations to check it out,

and over the last two hundred years,

many of the world's greatest mathematicians

have applied themselves to considering the problem

of how many of these symmetry groups there are.

Which really seems to have needed group theory.

So that needed Evariste Galois in the early 1800's

to see that algebraic equation permutations

could be worked with as groups,

after which Sophus Lie

(pronounced Lee)

found a way to do the same with differential groups,

groups of continual transformations.

And with that, the basic tools necessary for the classification of symmetry groups were in place,

at least enough for a base from which to search for new groups, anyway.

In the 1890's, Wilhelm Killing and Elie Cartan

managed to arrange the symmetry atoms in a table,

much like the periodic table, but of Lie groups,

with seven families A-G,

with up to nine ranks

related to how many dimensions they operate in,

and even though they may be impossible to picture,

these groups are real enough to have recognisable applications in quantum physics.

But these Lie groups were by nature infinite,

and it took Leonard Dickson at the start of the 20th century

to find a way to make them finite,

and geometrically he could only manage it for A to D groups.

however Jacques Tits in the 1950's

imagined a geometry of buildings formed from multi-crystals

living in a collection of intersecting universes,

each crystal an extension of one of the Platonic solids into a different dimension.

That hurried things along.

Then Richard Brauer came up with the idea of

an 'involution centralizer',

a kind of crystal cross section,

that Walter Feit and John Thompson proved could be used

to finally fully classify the individual symmetry groups,

but this seemed to throw up some discrepancies with the original table of Lie groups.

Then Zvonimir Janko showed

that the five exceptions to the four main families of Lie groups

were not the only outliers,

These oddities began to come to light in the 1960's

when John Conway used John Leech's 24 dimensional lattice

to discover yet more groups,

followed in the 1970's by Bernd Fischer

who discovered some huge groups,

the largest having 4,154,781,481, 226,426,191,177, 580,544,000,000 symmetries.

But even this ridiculously large number

was completely dwarfed by the size of the Monster.

In the end,

the vast number of calculations required for the construction of the Monster

needed newly invented computers to finalise its construction,

which turned out to be a group with an astonishing number of symmetries involved,

taking place in 196,883 dimensions.

So all that took place during the 160 years after Galois' interest in group theory,

and you might wonder what this has to do with anything

other than mathematics,

because even there it is an extraordinary oddity.

But it just so happens that John McKay noticed an even more extraordinary relationship

with something called the J-function in Number Theory.

which makes it seem that in fact this monster number

is involved in structures at the heart of what humans like to call reality,

and indeed it does seem to have links to calculations related to string theory.

When the numbers become greater

than estimates of the number of atoms in the universe,

we have to wonder which is real,

the atoms or the numbers.

Humans experience what is real by the way they individually understand what is real,

so they will always disagree on truth.

But there is indeed a truly real,

for one of God's Beautiful Names is The Real.

So as long as you search with sincerity

for what is truly real,

that means that your quest is for God,

and remember, the Monster may be extremely big,

but there is no doubt that God is Greater