Axis of Ordinary

The Minimal Atoms

On a flat plane, some transformations can move shapes around without stretching them at all. These are the distance-preserving motions (isometries): translations, rotations, reflections (and the slightly less famous glide reflection).

It turns out that reflections alone generate all of them. In fact, any plane isometry can be done with at most three reflections.

Step 0: Why triangles pin down motion

Take three points A, B, C that aren't collinear (they form a triangle). Then there cannot be two different points P != Q that have the same distances to all three.

Why? If a point X has equal distance to P and Q, then X lies on the perpendicular bisector of segment PQ. So if A, B, C were all equally far from P and Q, they'd all lie on that same bisector, meaning they'd be collinear. Contradiction.

So, a point is uniquely determined by its distance to three non-collinear points. That means: once you know where an isometry sends a triangle's three vertices, you know what it does to every point.

The 3-reflection construction

Suppose we have a blue triangle ABC and a red triangle A'B'C' of the same shape and size (congruent). We want a distance-preserving motion that maps the blue triangle onto the red one.

We'll build it using reflections only:

Reflection 1:

Reflect across the perpendicular bisector of segment AA'.
This sends A exactly to A'.

Reflection 2:

Now reflect across the perpendicular bisector of the current position of B and B'.
Crucial point: since distances are preserved and the triangles are congruent, A' is the same distance from those two points. So A' lies on that bisector, and is not moved by this reflection.

Result: B lands exactly on B' and A' stays put.

Reflection 3:

If C still doesn't match C', reflect across the perpendicular bisector of the current C and C'.
Again, because distances are preserved and now A' and B' already match, both A' and B' are equidistant from C and C', so they lie on the bisector line and don't move.

Result: C lands on C', with A', B' unchanged.

So we've matched the whole triangle using at most 3 reflections.

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This “minimal set generates a whole world” idea is everywhere:

1. In Boolean logic, you don't need {AND, OR, NOT}. A single operator like NAND can express everything.

2. In number theory, every integer >1 has a unique factorization into primes.

3. In signal processing, Fourier analysis builds signals out of sine waves.

4. In linear algebra, a basis generates a whole vector space.

5. Any permutation of n objects can be built from repeatedly swapping two items. And amazingly, for n >= 3, you can generate every shuffle using just two moves: "rotate everyone one step" plus "swap two items".

6. A CPU that can only subtract two numbers and jump to another step if the result is negative can still be fully programmable. Everything else (addition, loops, if-statements) can be built from that.

7. In human vision, the brain reduces the continuous spectrum to three cone cell types and reconstructs color from that.

Small toolkits. Infinite consequences.

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