Symmetries

Symmetries are quite literally everywhere

Flip a playing card, stand in front of a mirror, rotate a square. Nothing essential changes. Our eyes are tuned to this invariance; from mosque tilings and Baroque facades to corporate emblems (think Mercedes).

So, why is symmetry so satisfying? Its structure is preserved under change. You can rotate, reflect, translate it and the object will remain indistinguishable from before. The feeling of "same-ness despite motion" is exactly the fundamental seed of group theory.

Now, instead of staring at the objects, focus on the moves that don’t ruin its structure.

For a regular snowflake, these moves are six rotations and six reflections across its axes. If you compose any two legal moves together you get another legal move. If you undo a move, you land on a legal move again. There is an internal logic to all of these transformations which is the idea of a group. Think of it as a way to capture the essence of transformation.

Now bear with me as I define this.

What is a group?

Informally you can think of a group as a collection of actions. These actions must compose cleanly, there must be a way to do nothing, and every action must be undoable. That's it. Surprisingly little structure.

Formal definition if you are curious

Concrete examples (to anchor the abstraction)

Integers under addition :

• Operation:
• Identity: 0 (adding 0 changes nothing)
• Inverse: for , the inverse is (adding undoes )
• Closure and associativity hold as usual.

Full symmetries of the snowflake (rotations + reflections):

• Elements: 6 rotations + 6 reflections = 12 total.
• Operation: do one symmetry, then another (compose transformations).
• This is the dihedral group of order 12, commonly denoted —the symmetry group of the hexagon.

Non-examples (what fails the contract)

Natural numbers under subtraction : Not a group. is not in , so closure fails; inverses fail too.

Square matrices under multiplication without inverses: The set of all real matrices under multiplication is not a group, because some matrices cannot be inverted. Restrict to the invertible ones () and you have a group again.

Integers under division : Not a group. Closure fails immediately since, e.g., is not an integer; inverses also fail; and division isn't associative.

When we say the symmetries of an object form a group, we mean: take all of the structure-preserving transformations, compose them, include the identity, and include inverses. That is a group under composition.

Wallpaper Groups

Now that we have a base understanding of what a group is, let's look beyond isolated objects and see what happens when patterns extend infinitely across a surface.

Introducing: the Wallpaper Groups.

This is the mathematical classification of every possible way to tile the plane with a repeating motif while preserving its symmetry.

At first glance, it seems as if there should be infinitely many such patterns. After all, artists have been designing these ornate tilings for millennia. Yet, astonishingly, mathematicians have proven that there are exactly 17 distinct wallpaper groups.

Rotations in the plane can only occur at certain angles that fit neatly together, specifically , , , or . Other rotation angles, like , cannot fill a plane without leaving gaps.

When these allowable rotations are combined with reflections and translations, there are only 17 possible consistent arrangements. Each wallpaper group represents a distinct group of transformations.

In other words, the world's patterns—from Persian tiles to modern floor designs—all obey this elegant algebraic universe.

Play around with the wallpaper groups!

Rotations, Dimensions, and the Geometry of Space

We have classified flat, repeating patterns by their symmetries. Now, let's expand past this and look at multiple dimensions.

In two dimensions, the key symmetries are translations, rotations, reflections, and glide reflections. In three dimensions we focus on rotations. These rotations form a continuous symmetry group: do one turn after another, do nothing (turn ), and every turn can be undone by turning back by the same angle. We call this structure .

You actually interact with constantly even if you never call it by name: dice, polyhedral dice, spinning objects.

In 4D, a point needs four coordinates . Rotations still mean "turn without stretching," but now you can rotate e.g. the plane and plane independently.

Lattices, Translations, and the Kissing Number

The translation symmetries of a wallpaper form a grid, a lattice: all integer combinations of a few basic translation vectors. In 2D the translation lattice looks like ; in 3D, ; in general, in n dimensions.

Now for the classic geometric puzzle, the kissing number problem: how many unit spheres can touch one in the center without overlap?

In 2D, the answer is 6; in 3D, it's 12. In 4D there is more room and the maximum is 24. In 8D, a lattice called achieves a kissing number of 240. In 24D, the Leech lattice reaches a wild 196,560.

The exact kissing numbers are known only in a handful of dimensions: 1, 2, 3, 4, 8, and 24. Elsewhere we mostly just know bounds.

Conway Groups and Mathematical Discovery

Take the Leech lattice and ask: what are all the moves that leave this pattern perfectly unchanged? John Conway catalogued these symmetries and found three new sporadic simple groups , now called the Conway groups.

They arise by looking at all distance-preserving moves, modding out a sign flip, and then "pinning" certain lattice directions and keeping only symmetries that fix them.

The Monster

Most finite simple groups fall into infinite families. Sporadic groups are the weird outliers; there are 26 of them. The largest is the Monster group.

How big is the Monster?

Its order (number of elements) is:

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

That's about .

Its prime factorization is .

The Monster's smallest non-trivial representation lives in 196,883 dimensions, built via structures tied to the Leech lattice.

• 20 of the 26 sporadics live inside the Monster ("happy family").
• 6 sit outside ("pariahs").
• The Conway groups are part of the happy family.

Conclusion

The key takeaway is to think transformations, not objects. Symmetry gives a single rigorous story for patterns that otherwise look unrelated.