# How does one draw the following diagram: P1, P2, P3, P4, P5, and P6 are six parabolas in the plane, each congruent to the parabola y = (x^2)/16. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola P1 is tangent to P2, which is tangent to P3, which is tangent to P4, which is tangent to P5, which is tangent to P6, which is tangent to P1. There is no standard instrument for drawing parabolas, as a compass for circles. Although one can cut a parabola from cardboard, in which case they will need some openings in the template to see where the axis is located.

The equation of the circle is `x^2 + y^2 = R^2 , ` while the equations of the axes are `x = 0 , ` `y = +- sqrt ( 3 ) x , ` and `y = +- 1 / sqrt ( 3 ) x .`

The first parabola has the equation `y = R + x^2 / 16 . ` It is simple to reflect it over the x-axis to get `-y = R + x^2 / 16 , ` but to get other equations we need to perform at least one rotation, say by `pi / 3 .`

The formulas are `x = 1/2 x' - sqrt(3)/2 y' , ` `y = sqrt(3)/2 x' + 1/2 y' , ` so the equation of a rotated parabola is

`sqrt(3)/2 x' + 1/2 y' = R + 1/16 (1/2 x' - sqrt(3)/2 y')^2`

The other three parabolas can be obtained from this equation by changing x to -x, y to -y, and both.

A platform that can draw such equations can be seen in the link attached.