Function Repository Resource:

# SqrtSpace

Move back and forth from the squared space or square root space of an algebraic number field

Contributed by: Ed Pegg Jr
 ResourceFunction["SqrtSpace"][root,pts] while tracking signs, converts Cartesian pts2 to algebraic values in or converts those values back to Cartesian.

## Examples

### Basic Examples (2)

Using ϕ, GoldenRatio or Fibonacci’s rabbit constant, convert points to the algebraic number field and build the Fermat triangle:

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Using ψ, the supergolden ratio or Narayana’s cow constant, convert points to the algebraic number field and build the supergolden triangle:

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Convert back to the original points:

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### Properties and Relations (2)

Under "Neat Examples" in GeometricScene, there is a mysterious output after "Decompose a triangle into similar triangles":

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This triangle is in the algebraic number field / geometric space of where ρ is the plastic constant:

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Convert the points back to original values:

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A simple application of ToNumberField does not recognize the points as being in either or , but does recognize the values when they get squared:

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These values are algebraic numbers:

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The actual point is also a pair of algebraic numbers:

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The signs here happen to be positive, so taking the square root of the algebraic version does not require extra steps:

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### Neat Examples (3)

Convert 19 points from the algebraic number field of the plastic constant ρ into 3D coordinates:

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Find the distances between these points in terms of powers of :

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Plot out the points:

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## Requirements

Wolfram Language 11.3 (March 2018) or above

## Version History

• 1.0.0 – 27 February 2019