AI, Instinct, and the Crossroads of Human Reason
Friday, July 31, 2026
Cognition
Wednesday, June 11, 2025
Pythagoras symmetry and Invariance
Part 1: How the Pythagorean Theorem Becomes the Equation of a Circle
a2+b2=c2The horizontal side (a) runs along the x-axis. Its length is simply the x-coordinate of the far corner. Let's call this length x. The vertical side (b) runs along the y-axis. Its length is the y-coordinate of that same far corner. Let's call this length y. The hypotenuse (c) is now the straight-line distance from the origin (0, 0) to the point (x, y).
x2+y2=c2
A circle is the set of all points that are an equal distance from a central point.
From our work with Pythagoras, we know the distance from the origin (0, 0) to a point (x, y) is related by the formula
.x2+y2=(distance)2From the definition of a circle, we know that for every point on the circle, the distance from the center is constant and is called the radius, r.
x2+y2=r2Part 2: The Concept of Symmetry
Example: A butterfly, a human face (approximately). The Circle's Symmetry: A circle has infinite lines of reflectional symmetry. Any straight line that passes through the center of the circle is a line of symmetry. This is a remarkable amount of symmetry.
Example: A pinwheel, a starfish, a snowflake. A square has rotational symmetry of 90°, 180°, and 270°. The Circle's Symmetry: A circle has perfect rotational symmetry. You can rotate it by any angle around its center, and it is indistinguishable from its original position. This is the highest possible degree of rotational symmetry.
Part 3: The Concept of Invariance
Symmetry is the property of the object that looks the same after a transformation. Invariance is the property of an equation or quantity that does not change after a transformation.
What Changes? The coordinates of any specific point on the circle change. For example, if you have a circle of radius 5, the point (5, 0) is on the circle. If you rotate the circle by 90 degrees, that point moves to a new location: (0, 5). The x and y values have changed. What is Invariant (What Stays the Same)? The Shape: The circle itself looks identical. This is the symmetry. The Equation: The equation
remains true for the new point.x2+y2=r2Original point:
. The equation holds.52+02=25New point:
. The equation still holds.02+52=25The equation is invariant under rotation. This is the algebraic way of saying "the circle has rotational symmetry." The form of the equation doesn't care how you rotate it.x2+y2=r2
Pythagoras gives us a rule for distance:
.x2+y2=(distance)2A circle is a geometric object defined by a constant distance (radius). This constant distance leads to the circle's equation:
.x2+y2=r2This equation is invariant under rotation. This invariance is the algebraic reflection of the circle's perfect rotational symmetry.

