His most famous popular contribution was the
Menger sponge (mistakenly known as
Sierpinski's sponge), a three-dimensional version of the
Sierpiński carpet. It is also related to the
Cantor set. With
Arthur Cayley, Menger is considered one of the founders of
distance geometry; especially by having formalized definitions of the notions of
angle and of
curvature in terms of directly measurable
physical quantities, namely ratios of
distance values. The characteristic mathematical expressions appearing in those definitions are
Cayley–Menger determinants. He was an active participant of the
Vienna Circle, which had discussions in the 1920s on social science and philosophy. During that time, he published an influential result on the
St. Petersburg paradox with applications to the
utility theory in
economics; this result has since been criticised as fundamentally misleading. Later he contributed to the development of
game theory with
Oskar Morgenstern. Menger's work on
topology without points followed
Whitehead's point-free geometry's approach and used shrinking regions of the plane to simulate points. Menger was a founding member of the
Econometric Society. ==Legacy==