Googology Wiki
Googology Wiki

Steinhaus-Moser Notation is a notation created by Hugo Steinhaus, and extended by Leo Moser.[1] The formula is:

  • Triangle(n) = nn = Steinhaustriangle.svg
  • Square(n) = \(\boxed{n}\) = n inside n triangles
  • Circle(n) = ⓝ = n inside n squares

Triangle(n) would be graphically displayed by n inside a triangle, and the same for Square and Circle.

n in a triangle, triangle(n)

Leo Moser extends this notation with pentagons, hexagons, heptagons, octagons, etc., where n inside a x-gon is equal to n inside n (x - 1)-gons. Of course, circles are no longer used in this version, and are replaced by pentagons.

Matt Hudelson[2] defines a similar version like so:

  • n| = Line(n) = nn
  • n< = Wedge(n) = n followed by n lines
  • Triangle(n) = n followed by n wedges
  • Square(n) = n inside n triangles
  • etc.

Steinhaus-Moser notation is technically a fast iteration hierarchy with \(f_0(n) = n^n\). With this initial rule, \(f_m(n)\) is equal to n inside an m+3-gon.

n inside an n-gon is roughly \(f_\omega(n)\) in the fast-growing hierarchy.


  • Square(n) = Trianglen(n) = triangle(triangle(...triangle(n)...)) with n triangle()'s,
  • Triangle(triangle(n)) = (nn)(nn) = 2(nn)
  • Triangle3(n) = triangle(triangle(triangle(n))) = ((nn)(nn))((nn)(nn))


function polygon(n, level):
    if level == 3:
        return nn
    r := n
    repeat n times:
        r := polygon(r, level - 1)
    return r


See also