Algebra and Serendipity: The Beautiful Mathematics of vZome

Scott Vorthmann

What is vZome?

vZome is a software application for modeling in exact, discrete geometry.

Zometool is a physical modeling system using such geometry, and the original target of vZome.

Drawing Polygonal Geometric Figures

We use vectors to encode the parts of the figure.

Linear transformations (scaling, translation, and rotation) give us symmetry, similarity, and proportion.

[\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} a x + b y \\ c x + d y \end{matrix}]

What sorts of figures can we create?

Kinds of Numbers

Real numbers can represent anything, but computations can introduce roundoff error, and nothing is ever exact.

Integer arithmetic is exact, but limits our geometry considerably.

There is an alternative: number systems that we can use exactly, but still give more expressiveness than integers.

The Golden Ratio

φ = \frac{1 + \sqrt{5}}{2} ≃ 1.618034

Solution to

x^{2} = x + 1

, so

φ^{2} = φ + 1

[1, φ^{2}, 0] = [(1 + 0 φ), (1 + 1 φ), (0 + 0 φ)]

All our coordinate values live in:

{a + b φ; a, b \in ℤ}

Golden Numbers

{a + b φ; a, b \in ℤ} = ℤ [φ]

, the Golden Ring

{q + r φ; q, r \in ℚ} = ℚ [φ]

, the Golden Field

(a + b φ) (c + d φ)

= a c + b c φ + a d φ + b d φ^{2}

= a c + b c φ + a d φ + b d + b d φ

= (a c + b d) + (b c + a d + b d) φ

\in ℤ [φ]

Rings can be Rich!

\frac{1}{φ} = φ - 1 \in ℤ [φ]

No general division (inverses), but still an infinite ring of inverses.

Exact scaling, up and down!

Still missing: general centroids, intersections, projections, and some inverse transformations.

The Joy of Exactness

Using $ℤ$ (or $ℚ$ ) means all arithmetic is exact; there is no roundoff error.

Equality testing is exact, with no need for "close enough" tests.

The scale of a model is limited only by the integer representation.

A connected path of line segments closes, or not, with no ambiguity.

The Power of $ℤ [φ]$

Scale up or down as far as we like, exactly

Nice proportions (triangles & rectangles) in ${ℤ [φ]}^{2}$

Penrose tiles in ${ℤ [φ]}^{2*}$

Icosahedral (H3) symmetry in ${ℤ [φ]}^{3}$

H4 symmetry and Icosians in ${ℤ [φ]}^{4}$

Quasicrystals

Rings, Fields, and Symmetry in vZome

Let's look at how these concepts manifest in vZome.

Octahedral Symmetry

Possible even in $ℤ^{3}$ , by permuting coordinates and flipping signs

Analogues of the octahedron and cube in any dimension

Peter Pearce, Synestructics SuperStructures

OK, What Else?

Are there other geometric rings or fields?

Can we represent other symmetry groups?

Coxeter: no more polyhedral groups

$ℤ [\sqrt{2}]$

Solution to

x^{2} = 2

(a + b \sqrt{2}) (c + d \sqrt{2})

= (a c + 2 b d) + (b c + a d) \sqrt{2}

| (\sqrt{2}, \sqrt{2}) | = | (1, 0) |

$ℤ [\sqrt{3}]$

Solution to

x^{2} = 3

(a + b \sqrt{3}) (c + d \sqrt{3})

= (a c + 3 b d) + (b c + a d) \sqrt{3}

the dodecagon in $ℤ [\sqrt{3}]$

$ℤ [\sqrt{3}]$ Tilings

Stampfli's 12-fold without squares

Square-triangle

Search here for "12-fold"

The Hat and Spectre both fit in this ring!

Polygon Rings

Peter Steinbach described an infinite class of rings derived from regular polygons.

Generally, these rings work nicely for substitution tilings.

We've seen the rings associated with the pentagon, the octagon, and the dodecagon.

Heptagon Numbers

ρ σ = ρ + σ ρ^{2} = 1 + σ σ^{2} = 1 + ρ + σ

\frac{1}{σ} = σ - ρ \frac{1}{ρ} = 1 + ρ - σ

The Heptagon Ring

ℤ [ρ, σ] = {a + b ρ + c σ; a, b, c \in ℤ}

compare with:

ℤ [\sqrt{2}, \sqrt{3}] =

{a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}; a, b, c, d \in ℤ}

The Plastic Number

ρ = 1.32471...

Solution to

x^{3} = x + 1

{a + b ρ + c ρ^{2}; a, b, c \in ℤ}

Wikipedia article

My iPython notebook

Algebra and Serendipity: The Beautiful Mathematics of vZome

Scott Vorthmann

What is vZome?

Drawing Polygonal Geometric Figures

Kinds of Numbers

The Golden Ratio

Golden Numbers

Rings can be Rich!

The Joy of Exactness

The Power of $ℤ [φ]$

Rings, Fields, and Symmetry in vZome

Octahedral Symmetry

OK, What Else?

$ℤ [\sqrt{2}]$

Capabilities of $ℤ [\sqrt{2}]$

$ℤ [\sqrt{2}]$ Tilings

$ℤ [\sqrt{3}]$

$ℤ [\sqrt{3}]$ Tilings

Polygon Rings

Heptagon Numbers

The Heptagon Ring

Heptagon Explorations

The Plastic Number

Algebra and Serendipity: The Beautiful Mathematics of vZome

Scott Vorthmann

What is vZome?

Drawing Polygonal Geometric Figures

Kinds of Numbers

The Golden Ratio

Golden Numbers

Rings can be Rich!

The Joy of Exactness

The Power of ℤ[φ]

Rings, Fields, and Symmetry in vZome

Octahedral Symmetry

OK, What Else?

ℤ[2]

Capabilities of ℤ[2]

ℤ[2] Tilings

ℤ[3]

ℤ[3] Tilings

Polygon Rings

Heptagon Numbers

The Heptagon Ring

Heptagon Explorations

The Plastic Number

The Power of $ℤ [φ]$

$ℤ [\sqrt{2}]$

Capabilities of $ℤ [\sqrt{2}]$

$ℤ [\sqrt{2}]$ Tilings

$ℤ [\sqrt{3}]$

$ℤ [\sqrt{3}]$ Tilings