Overview
Interactive HermitesProof tool for mathematics exploration
Hermite's Proof
Interactive Hermite's Proof visualization and calculation tool.
Hermite's Problem: Cubic Irrational Periodicity
Input Controls
Example Numbers
Cubic Irrationals
Quadratic Irrationals
Rational Numbers
Transcendental Numbers
Step-by-Step Solution
Step 1: Initial Setup
We represent the value α = 2.154435 in projective space using a triple:
Why this representation? Using the triple (α, α², 1) allows us to work in projective space, where cubic irrationals exhibit periodicity in their transformations. This is the key insight that solves Hermite's problem—cubic irrationals have a unique "signature" in their projective behavior that distinguishes them from other numbers.
Step 2: HAPD Algorithm
Algorithm Steps
Mathematical Significance
The HAPD algorithm extends the continued fraction idea to 3D projective space, allowing us to detect periodicity in cubic irrationals.
Integer extraction: Similar to continued fractions, we extract integer parts from each coordinate ratio.
Projective transformation: The algorithm performs a special transformation that preserves the algebraic properties of cubic irrationals.
Periodicity detection: Cubic irrationals will eventually repeat their position in projective space, creating a cycle.
Step 3: Matrix Verification
Step 4: Connecting the Methods
The HAPD algorithm and matrix verification provide two complementary ways to identify cubic irrationals:
Projective Approach (HAPD)
- Works in projective space with the triple (α, α², 1)
- Detects periodicity through direct computation
- Shows the cyclic nature visually through trajectory
- Related to continued fraction expansions
Algebraic Approach (Matrix)
- Works with the companion matrix of the minimal polynomial
- Verifies through trace relations
- Connects to eigenvalues and characteristic polynomials
- Provides algebraic confirmation
The Unified View
Both methods are mathematically equivalent but offer different insights. The HAPD algorithm provides a concrete computational procedure, while the matrix approach gives us the abstract algebraic justification. Together, they form a complete proof that periodicity in the algorithm is a perfect characterization of cubic irrationals, just as periodicity in continued fractions characterizes quadratic irrationals.
This dual verification confirms that for α = 2.154435, the observed behavior is not coincidental but a fundamental mathematical property, solving Hermite's 170-year-old problem.
Enter a value and click "Compute" to start