Pythagoras symmetry and Invariance

 

Part 1: How the Pythagorean Theorem Becomes the Equation of a Circle

This is a beautiful and direct application of a familiar theorem in a new context.

1. Start with Pythagoras

You know the Pythagorean Theorem: For any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, c) is equal to the sum of the squares of the other two sides (a and b).

It is interesting to note that a + b != c.  But mathematicians look for other options. This must be trial and error of finding what happens if squares of the numbers are added.

$a^2+b^2=c^2$

2. Place it on a Coordinate Plane

Now, let's draw this triangle on a Cartesian (x-y) graph. To make it simple, we'll place the right-angle corner at the origin (0, 0).

• The 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).

If we substitute our new labels (x, y) into the Pythagorean theorem, we get:

$x^2+y^2=c^2$

This equation gives us the distance (c) from the origin to any single point (x, y).

3. Define a Circle

Now, what is the geometric definition of a circle?

A circle is the set of all points that are an equal distance from a central point.

That "equal distance" is what we call the radius (r).

4. Combine the Ideas

Let's put the center of our circle at the origin (0, 0). We want to find the equation that describes all the points (x, y) on the edge of this circle.

• From our work with Pythagoras, we know the distance from the origin (0, 0) to a point (x, y) is related by the formula 

  $x^2+y^2=(\text{distance}{)}^2$

  .

• From 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.

So, we just replace the "distance" (c) in our Pythagorean formula with the "radius" (r):

$x^2+y^2=r^2$

This is the equation of a circle with its center at the origin. It's not just describing one triangle anymore; it's a rule that describes the infinite number of possible right-angled triangles you can draw to any point on the circle's circumference.

In summary: The Pythagorean theorem gives you the distance between two points. The equation of a circle is simply a statement that all points on the circle must satisfy the Pythagorean theorem for a fixed distance (the radius) from the center.

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Part 2: The Concept of Symmetry

Symmetry is a concept of balance, harmony, and "sameness." In mathematics, we formalize this: an object is symmetric if it looks the same after a certain operation, called a transformation, is applied to it.

Let's look at the main types, using the circle as our main example.

1. Reflectional Symmetry (Bilateral Symmetry)

This is the "mirror image" symmetry. An object has reflectional symmetry if you can draw a line through it (a "line of symmetry") and the object is a perfect mirror image of itself on either side of the line.

• 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.

2. Rotational Symmetry

An object has rotational symmetry if you can rotate it around a central point by less than a full 360 degrees and it looks identical to how it started.

• 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.

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Part 3: The Concept of Invariance

Invariance is the other side of the symmetry coin.

• 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.

If an object has a symmetry, its description (its equation) must have a corresponding invariance. Let's tie this directly to the circle.

The Transformation: Let's take the circle and rotate it around its center.

• 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)?

  1. The Shape: The circle itself looks identical. This is the symmetry.

  2. The Equation: The equation 

     $x^2+y^2=r^2$

      remains true for the new point.

     • Original point: 

       $5^2+0^2=25$

       . The equation holds.

     • New point: 

       $0^2+5^2=25$

       . The equation still holds.
       The equation 

       $x^2+y^2=r^2$

        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.

The Big Picture Connection

• Pythagoras gives us a rule for distance: 

  $x^2+y^2=(\text{distance}{)}^2$

  .

• A circle is a geometric object defined by a constant distance (radius).

• This constant distance leads to the circle's equation: 

  $x^2+y^2=r^2$

  .

• This equation is invariant under rotation.

• This invariance is the algebraic reflection of the circle's perfect rotational symmetry.

Invariance is one of the most powerful ideas in science. In physics, the laws of nature are believed to be "invariant" under certain transformations. For example, the laws of physics are the same today as they were yesterday (invariance under time translation) and the same in New York as in Tokyo (invariance under spatial translation). These symmetries are what lead to fundamental laws like the conservation of energy and momentum.