In addition to his contributions to combinatorial design theory, Jakob Steiner was one of the greatest geometers of all time.
As previously discussed, the «schoolgirl problem» has seven non-isomorphic solutions, listed in 1922 by American mathematician Frank Nelson Cole. Cole, known for discovering the factors of the 67th Mersenne number (267 – 1) in the early 20th century, achieved this feat using only paper and pencil, dedicating every Sunday for three years to the task. In addition to his contributions to combinatorial design theory, Swiss mathematician Jakob Steiner, renowned for his ‘minimum trees‘ or graph ‘bonsais‘, was one of the greatest geometers of all time, second only to Apollonius of Perga according to some. Steiner, a staunch critic of analytic geometry, exclusively used synthetic and projective geometry methods in his work. He is best known for his research on conics, particularly his circumscribed and inscribed ellipses in a triangle. Steiner’s circumscribed ellipse, or ‘circumellipse‘, is the only ellipse that passes through the three vertices of a triangle, with its center at the triangle’s centroid. Contrary to what some might think, a circumscribed circle is not a Steiner circumellipse because its center is the intersection point of the triangle’s bisectors, not its medians. Among its properties, the Steiner circumellipse has the smallest area of all ellipses circumscribed to a triangle. Conversely, the Steiner inscribed ellipse, or ‘inellipse‘, is the only ellipse inscribed in a triangle that is tangent to the midpoints of its sides. The surface of the Steiner inellipse is a quarter that of the circumellipse. Readers are encouraged to start their analysis with the simpler case of a circumellipse and an inellipse of an equilateral triangle.
