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Three of my favorite ellipses
Top: Eccentricity = .5. Semi major axis and focal points suggest two equilateral triangles.
Middle: Eccentricity = ~.618, aka 1/phi. Phi = ~1.618. Also known as the golden ratio. The perpendiculars through the focal points suggest a square.
Bottom: Eccentricity = square root of 1/2. Semi major axis and focal points suggest a square.
Top: Eccentricity = .5. Semi major axis and focal points suggest two equilateral triangles.
Middle: Eccentricity = ~.618, aka 1/phi. Phi = ~1.618. Also known as the golden ratio. The perpendiculars through the focal points suggest a square.
Bottom: Eccentricity = square root of 1/2. Semi major axis and focal points suggest a square.
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Very interesting! Do you know if there are generalisations of the properties for more sided polygons, and which eccentricities appear?