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Steinhaus-Moser Notation is a notation created by Hugo Steinhaus, and believed to be extended by Leo Moser.[1]


Steinhaus defined the notation in Mathematical Snapshots as[2]

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

n in a triangle, triangle(n)

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

Steinhaus also defined Mega = ② and Megiston = ⑩ with this notation.

It is believed that Leo Moser extended this notation with pentagons, hexagons, heptagons, octagons, etc., where n inside a x-gon is equal to n inside n (x - 1)-gons, although it is unknown if and where Moser actually made such extention. Of course, circles are no longer used in this version, and are replaced by pentagons.

Hudelson's notation[]

Matt Hudelson[3] 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.

Susan's notation[]

n inside an m-gon is written as n[m] with Susan's general notation.[4]

For example, a[3] is a inside triangle, a[4] is a inside square. a[3][3] is a inside 2 triangles, and it can also be written as a[3]2 by Susan's notation.


  • 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))
Exact results of x inside n triangles
Number of triangles (n) Value
0 x
1 xx
2 \((x^{x})^{(x^{x})}\) = \(x^{x^{(x+1)}}\)
3 \(x^{x^{(x^{x+1}+x+1)}}\)
4 \(x^{x^{(x^{(x^{x+1}+x+1)}+x^{x+1}+x+1)}}\)

A pattern can be attained from this: For any number of the form \(a^{b^{c}}\), the result of placing it in a triangle will be \(a^{b^{b^{c}+c}}\).


Leonardıs et al. (2022) proved[5]

\(n\uparrow \uparrow (n+1) \le n[4] \le n \uparrow n \uparrow (n+1) \uparrow\uparrow (n-1) \le n \uparrow\uparrow (n+2)\)

and also

\(n\uparrow \uparrow \uparrow (n+1) \le n[5] \le n \uparrow \uparrow (n+1) \uparrow\uparrow\uparrow n \)

Therefore it is clear that

\(n\uparrow \uparrow \uparrow (n+1) \le n[5] < (n+1) \uparrow\uparrow\uparrow (n+1) \)

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.


The Jain work Tiloyapaṇṇatti (ca. 400 AD) likely specifies the operation "vargita-saṃvargita" (or V.S) as a number raised to its own power[6]. The work then goes on to define various large numbers by applying this operation to a number multiple times, equivalent to a number inside multiple triangles.

The work also defines the operation "śalakātrayaniṣṭhāpana" (or S.T) as applying V.S to a number, that number of times, then applying V.S to that number that number of times again, and then one more time applying V.S to that number, that number of times. This is equivalent to a number inside 3 squares, or approximately \(x\uparrow \uparrow \uparrow 4\). This means that if the Tiloyapaṇṇatti is the true source of V.S and S.T, then the ancient Indians had developed an early form of Steinhaus-Moser notation around 1500 years before it was defined by Steinhaus and Moser.


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


  1. Steinhaus-Moser Notation
  2. Hugo Steinhaus. Mathematical Snapshots Courier Corporation, 1999. ISBN 9780486409146 p.28
  3. [1]
  4. Susan Stepney Moser's polygon notation Retrieved 2023-02-10
  5. Leonardıs, A., D'atrı, G. & Caldarola, F. (2022). Beyond Knuth's notation for unimaginable numbers within computational number theory. International Electronic Journal of Algebra, 31 (31), 55-73 . (Preprint)
  6. Demonin2 Pentation in Ancient Indian Mathematics (śalakātrayaniṣṭhāpana) 2023-09-02

See also[]