A pretty picture involving the Gaussian Integral

The Gaussian integral is a neat and very easy representation of the bell curve with the area under the curve exactly equal to the square root of pi.  Nice simple-looking expression.  The usual bell curve formula looks a lot worse since it's a family of curves and it also sets the area under the curve equal to one instead of the square root of pi.  Or something of that nature.

Inspired by that expression, I used Desmos to plot the picture below.  The darker shape is just a semi-circle with radius equal to the fraction one over the 4th root of pi.  But I designed the plot so that the dark semi-circle area in the picture was EXACTLY equal to the light orange area between the semi-circle and the bell curve.  I think that precise relationship makes a geometric picture such as this look quite nice, even though it's difficult to intuitively tell the areas are indeed equivalent.  The picture kind of looks like a pretty sunset if you fill it in with the right colors.  Both areas should be equal to the square root of pi over two if I didn't make any errors.

And here are the expressions I typed into Desmos to generate the aesthetically pleasing plots shown above.

I think that's enough fun for one day.