cursed equations

What if we put shapes into equations? This page is the result of my journey to answer that question.

We view a 'shape' as a map from the interval [0,1] to the complex plane. I like to think of this as drawing the shape - the point where we place our pen is A(0), and as we draw, the input increases until we reach the end of the shape at A(1). Using this definition, these shapes can be operated on, by operating the outputs of their maps. For example: the shape A+B is defined by the map (A+B)(x) = A(x)+B(x)

I first explored this idea of operating shapes in a video about the quadratic formula . This formula is a great demonstration because it includes all basic arithmetic operations, as well as a square root. Later I continued this exploration with a video on the Pythagorean theorem - both the typical 2D version and its 3D counterpart.

navigation

The primary interaction with this site is to adjust the equation and the shapes.

The current equation is displayed in the lower righttop center. Use the arrows on the left and right to try out different equations.

There are 2 or 3 shapes to adjust, depending on the equation. Click on a, b, or c to select the shape to edit. Then use the arrows to adjust its properties:

• shape: the number of vertices, or a special definition like 'spiral'
• step: the total 'laps' the path makes around the center of the shape
• animation: the transformation applied to the shape
• type: the path taken between vertices, if using a regular polygon

Use the icons to adjust other settings, such as toggling the grid or points. The 'detail' icon controls the level of detail of all shapes. Setting the dial to the left will be low quality and fast, while the right will be high quality and slow.

about me

I am TheGrayCuber, a creator of mathematical websites and videos!

This concept of operating shapes began while reading Galois Theory by Ian Stewart, where I encountered a proof of the Fundamental Theorem of Algebra. The proof uses a circle as the input of a polynomial. After making a video about the proof, I was inspired to try taking a circle as an input into other equations, such as sine and cosine.

While exploring these functions on a circle, I wondered about other shapes. What would they look like when put through a function? This lead to the definition given above - a map from [0,1] - which allowed for a lot of fun with various shapes!

Since this project, I have experiemented with this 'shape algebra' in other systems beyond the complex numbers. Check out my hypercomplex grapher to explore the behavior in some weird systems.

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