MH

 A

The Hexagrammum Mysticum (Mystic Hexagram) is the spectacular core theorem that Blaise Pascal discovered in 1639 when he was just 16 years old.

It is a fundamental theorem in projective geometry. What makes it "mystic" isn't magic, but rather the sheer, mind-boggling complexity of interconnected alignments that naturally emerge from a simple geometric setup.

The Core Theorem

In its simplest modern phrasing, the theorem states:

If you choose any six points on a conic section (such as an ellipse, circle, parabola, or hyperbola) and connect them in any order with straight lines to form a six-sided figure (a hexagon), the three pairs of opposite sides will intersect at three points that lie on a single straight line.

Pascal's Theorem in action. Source: Wikipedia

How to trace it in the visual above:

1. The Shape: Look at the six points on the ellipse labeled A,B,C,D,E,F. They are connected to form a self-intersecting hexagon.

2. Opposite Sides: The pairs of "opposite" sides are color-coded:

   • Line AB and Line DE (red) intersect at point G.

   • Line BC and Line EF (blue) intersect at point K.

   • Line CD and Line FA (yellow) intersect at point H.

3. The Pascal Line: Notice how the three newly created white intersection points (G, H, and K) align perfectly. You can draw a single straight white line right through all of them. This is called the Pascal Line.

Why is it called a "Mystic" Hexagram?

The magic happens when you realize you can connect those six initial points in any order you want.

Because you have 6 points, there are $6!/2\cdot 6=60$ different ways to connect them into a hexagon. Every single one of those 60 variations produces its own unique Pascal Line.

When mathematicians mapped out how these 60 lines interact, they discovered a beautifully intricate, web-like network of symmetric intersections:

• The 60 Pascal Lines intersect in groups of three at 20 distinct points (called Steiner points).

• They also intersect in another set of 60 points (called Kirkman points).

• The Steiner points themselves align beautifully on 15 new lines (called Plücker lines).

Pascal reportedly proved all of these cascading properties in his original treatise, The Geometry of Conics. Sadly, the full manuscript was lost to history, leaving only his brief 1640 promotional broadsheet (Essay pour les coniques) behind to tease the mathematical community.