The Hyper Moser (and other Mega Numbers) - Numberphile

Key Concepts

• Triangle Notation: n inside a triangle means n to the power of n (n^n).
• Square Notation: n inside a square means n nested triangles, each containing n.
• Circle Notation: n inside a circle means n nested squares, each containing n.
• Omega: Defined as Circle 2, a very large number.
• Double Arrow Notation (Knuth's Arrow Notation): Represents repeated exponentiation or a power tower. a double arrow b means a to the a to the a... b times.
• Pentagon Notation (Moser's Notation): n inside a pentagon means n nested squares, each containing n.
• Auga: Defined as Hexagon 2, an even larger number than Omega.
• n-gon Notation: A generalization where an (n+1)-gon of n is n nested n-gons, each containing n.
• Mosa: Defined as a megagon (a polygon with Omega sides) of 2, an extremely large number.
• Grand Mosa: A megagon of 3.
• Great Mosa: A megagon of 4.
• Super Mosa: A mosagon (a polygon with Mosa sides) of 2.
• Super Super Mosa: A super mosagon of 2.
• Hyper Mosa: A super^Mosa mosagon of 2 (Mosa number of "super" prefixes).

Steinhaus's Notation and Omega

• Triangle Notation: The video begins by explaining Hugo Steinhaus's notation for generating large numbers, starting with the triangle. Triangle(n) = n^n.
  • Example: Triangle(2) = 2^2 = 4, Triangle(3) = 3^3 = 27, Triangle(4) = 4^4 = 256.
• Square Notation: Square(n) is defined as n inside n triangles. This means applying the triangle function n times.
  • Example: Square(2) = 2 inside 2 triangles = Triangle(Triangle(2)) = Triangle(4) = 4^4 = 256.
  • Square(3) = 3 inside 3 triangles = Triangle(Triangle(Triangle(3))) = Triangle(Triangle(27)) = Triangle(27^27) = (27^27)^(27^27).
• Circle Notation: Circle(n) is defined as n inside n squares.
• Omega: Omega is defined as Circle(2). This is 2 inside 2 squares.
  • Omega = Square(Square(2)) = Square(256). This means 256 inside 256 triangles.
  • Omega is bounded by 2↑↑259 < Omega < 2↑↑260, where ↑↑ represents Knuth's arrow notation (power tower).
• Comparison to Google and Googleplex:
  • Google (< 2↑↑5)
  • Googleplex (< 2↑↑6)
  • Omega is vastly larger than both.
• Properties of Omega:
  • Its full value cannot be written out.
  • Algorithms exist to calculate its last digits.
  • The last digits of Omega are known to be ...56.
  • It is speculated to begin with a 1.
• Related Numbers:
  • Triton: Circle(3)
  • Magiston: Circle(10)

Moser's Notation and Mosa

• Pentagon Notation: Leo Moser extended Steinhaus's notation by introducing the pentagon. Pentagon(n) is defined as n inside n squares.
• Auga: Auga is defined as Hexagon(2), which is 2 inside 2 pentagons. This is equivalent to Pentagon(Omega).
• General n-gon Notation: An (n+1)-gon of n is defined as n inside n n-gons.
• Mosa: Mosa is defined as a megagon (a polygon with Omega sides) of 2. This is an incredibly large number.
• Historical Context: The concept of building large numbers in this way dates back 1500 years to ancient Indian mathematics (Varita Sagita).
• Properties of Mosa:
  • It is an even number.
  • Its last digits are known to be ...8030156.
  • It is bounded by 2↑^(Omega-2) 3 < Mosa < 2↑^(Omega-2) 4, where ↑^(Omega-2) represents Omega-2 arrows.
• Comparison to Graham's Number: Mosa is smaller than Graham's number. This is because the number of arrows in the Knuth's arrow notation representation of Graham's number grows much faster.
  • G1 (first stage in Graham's number construction) = 3↑↑↑↑3, which is already much larger than Omega.
  • Since G2 has G1 arrows, it is larger than Mosa, and therefore Graham's number (G64) is also larger.

Beyond Mosa: Pushing the Limits

• Grand Mosa: A megagon of 3.
• Great Mosa: A megagon of 4.
• Super Mosa: A mosagon (a polygon with Mosa sides) of 2.
• Super Super Mosa: A super mosagon of 2.
• Hyper Mosa: A super^Mosa mosagon of 2 (Mosa number of "super" prefixes). This is an extremely large number.
• Computational Limits: The video emphasizes that these numbers are so large that they cannot be calculated or comprehended within the observable universe.

Conclusion

The video explores the fascinating world of large numbers, starting with Steinhaus's notation and culminating in the mind-boggling concept of the Hyper Mosa. It demonstrates how mathematicians have devised ingenious methods to generate numbers far beyond human comprehension, highlighting the power of recursive definitions and the limitations of our ability to grasp the truly infinite. While these numbers are largely theoretical, they provide a glimpse into the abstract and boundless nature of mathematics.