**The Math**

**35+ years of applied hyperbolic 3-manifold geometry**

Why Hyperbolic Geometry?

### this non-Euclidean math most accurately explains our non-Euclidean complex adaptive universe

# Continuing a Legacy of Hyperbolic Geometry Innovation

## Hyperbolic geometry conceptualized

January 1, 1823

proving that the 2,000 year old Euclidean geometry is not the only representation of the world.

## Non-Euclidean geometry believed to represent space more accurately than Euclidean

November 1, 1859

Riemann is the first documented application of non-Euclidean geometries to represent physical reality.

## Poincare’s Conjecture creates a seemingly unsolvable problem for hyperbolic geometry

January 1, 1904

leaving many mathematicians in argument or just straight confused as it couldn’t be proven true for all dimensions

## Einstein uses non-Euclidean geometry to conceptualize a curved spacetime in his general theory of relativity

November 1, 1915

shifting the paradigm of reality from flat, Euclidean, and absolute to curved, non-Euclidean, and relative as well as popularizing the notion of a 4th dimension

## relativity carried forward to understand complex adaptive systems

January 1, 1940

systems theory applies relativity and nonlinear relationships, sparking new theories which accurately predict phenomena such as the butterfly effect. However, due to the n-body problem

## William (Bill) Thurston creates the Geometrization Conjecture

May 1, 1982

bring the math community to consensus on the trueness of Poincare’s conjecture. This reignited hyperbolic geometry with a new generation of thinkers seeking to prove

## Bill receives the Fields Medal

August 1, 1982

equivalent to the Nobel Prize for mathematicians under 40 years old, for revolutionizing the study of topology in 2 and 3 dimensions, showing interplay between

## Robert Meyerhoff discovers smallest of infinitely many hyperbolic manifolds

July 18, 1987

discovers a lower bound for the volume of hyperbolic 3-manifolds as a graduate student under Bill’s guidance, solving one of the problems outlined by Thurston’s

## David Gabai begins proving more parts of Thurston’s conjecture

July 7, 1989

as a graduate student under the guidance of Bill. His results played a key role in proving core concepts of topology of 3-manifolds including essential

## Nathaniel Thurston, Gabai, and Meyerhoff apply code to hyperbolic 3-manifold geometry

September 13, 1996

Bill Thurston’s son carries on the legacy, pioneering the use of computers to help prove things like the theorized smallest hyperbolic 3-manifold. This seminal work was

## Gabai announced chairman of Princeton’s math department

August 17, 2012

due to his contributions in collaboration with Nathaniel and guided by Bill’s conjecture

## Meyerhoff announced chairman of Boston College’s math department

August 1, 2013

due to his contributions in collaboration with Nathaniel and discoveries guided by Bill’s conjecture

## Nathaniel cofounds ipvive to design the first AI based on hyperbolic 3-manifolds

January 1, 2014

leaving Google’s Rank Labs team due to his belief that hyperbolic geometry, not Euclidean, is necessary to create an unbiased AI capable of ultra-personalization. Hyperbolic

## Hyperbolic 3-Manifold AI is the first AI to learn and play BRIDGE

September 1, 2020

learns each player’s lens (using hyperbolic 3-manifold lenses) from natural communications and logical gameplay logic in order to understand the relationships between team players and