Encyclopedia/All topics/Golden ratio

Golden ratio

Articles 1–9 of 9. Alphabetical by title.

• Phi Support Alternatives

  The golden ratio is the only positive number that solves x² = x + 1, and a machine-checked proof confirms that e, π, and square roots all fail the test.

• Phi Support Alternatives Common Constants Fail Selection

  The golden ratio is the only positive number that solves x² = x + 1, and a machine-checked proof confirms that e, π, and three square roots do not.

• Phi Support Alternatives Sqrt2 Fails Selection

  The golden ratio is the positive number that solves x² = x + 1; a machine-checked proof shows √2 fails that test.

• Phi Support Alternatives Sqrt3 Fails Selection

  The golden ratio is not picked from a menu of pretty constants; a simple equation rules every rival out.

• Phi Support Alternatives Sqrt5 Fails Selection

  The golden ratio φ is famously built from √5, but the number √5 itself fails the one equation that selects φ.

• Phi Support Lemmas

  The golden ratio is the unique positive number that equals one plus its own reciprocal, a fact that anchors the framework's self-similarity arguments.

• Phi Support Lemmas Exclusivity Model Independent

  The golden ratio is the one positive number that solves x² = x + 1; a machine-checked theorem proves no other positive number can.

• Phi Support Lemmas One Lt Phi

  The golden ratio is greater than 1, a fact so basic it underpins the entire Recognition Science framework's claims about scale and structure.

• Phi Support Lemmas Phi Ne Zero

  The golden ratio, defined as (1+√5)/2, is a positive number, and a machine-checked proof records that fact so later steps can safely divide by it.