The GEOMETOR project is motivated by our sense of wonder

over a simple proportional relationship.

- exploring the architecture of all that is

- seeking to understand the ordering principles of the universe

**Seeking** Truth Through Divine Logic

Exploring The Architecture of All That Is

- create a platform for exploration

- collaboration
- education
- discovery

**GEOMETOR Explorer** is an essential tool of the GEOMETOR Project.

The purpose of **Explorer** is to find and catalogue unique instances of golden sections within geometric constructions with algebraic proofs.

**Explorer** is currently a demonstration prototype. There are significant plans for enabling user interactivity including adding lines and circles within the interface, as well as analysis of the discovered golden sections.

check out **Explorer** here:

these two can take up to a few minutes to load

only tested with Chrome and Firefox - Chrome is significantly faster than Firefox.

**Explorer** operates under the rules of classical constructive geometry within a planar field. Explorer provides two virtual tools for constructions - an unmarked straight edge and a compass. Both of these tools require two points to operate. We align the straightedge with two points to trace a line. We align the compass with two points to trace a circle. This simple framework for the study of geometric proportions has been with us since the ancients and is the foundation of Euclid's Elements.

Every geometric construction within **Explorer** begins with a blank slate. The only "givens" are the first two **starting points** on our field. Everything else must be constructed.

The distance between these two starting points represents the unit measure of the field - a distance of one. All other constructions are expressed in proportion to this unit.

The intersections of lines and circles identify new points on the field - creating opportunities for more lines and circles - and of course, more intersection points.

Line **segments** and circular **sectors** and **arcs** are used within **Explorer** as graphical illustrations and are not used directly in constructions.

Lines and circles are proportions that extend from their initial points without end. Therefore, lines are always extended beyond the screen and circles are always drawn in full. By expressing the elements fully, we allow for more discovery of relationships and intersections.

While following the formality of Euclid's constructive geometry, **Explorer** incorporates two concepts that were undeveloped in Euclid's time: the cartesian grid and algebra.

With our unit measure established by the starting points and a notion of perpendicularity, we establish a horizontal (x) and vertical (y) scale. We use these scales to identify the position of points as `[x y]`

.

The origin is the point half-way between the starting points. In the cartesian plane this is `[0 0]`

.

As our givens, the starting points are the only points without parents and are placed on the field with positions of `[1/2 0]`

and `[-1/2 0]`

All other points are derived algebraically from the intersection of elements.

Decimal equivalents are for applied mathematics. All calculations within

Explorerare algebraic.

Constructions within **Explorer** are currently scripted. Lines and Circles are created by passing Point references as parameters.

Explorer maintains an array of constructed points and elements.

Here is the script that constructs the current geometry...

]]>A collection of articles

]]>- divine proportion
- golden ratio
- extreme and mean ratio

The Golden Ratio is a proportion of two unequal values a and b where

\[ \large \frac{a}{b} = \frac{b}{a+b}\]

in other words - two unequal values a and b where \(\frac{a}{b} = \frac{b}{a+b}\)

The Golden Ratio emerges at the intersection of proportional systems

exists in the interference pattern of elements

and in nature

]]>\[ \large \mathbf{\Phi^2 \, - \, \Phi \, - \, 1= \, 0 }\\\]

\[ \large \mathbf{roots\; of\; \phi \, := \, \left\{ \left(\frac{-\sqrt{5} + 1}{2}, 0 \right), \left(\frac{\sqrt{5} + 1}{2}, 0 \right) \right\} }\]

\[ \large \mathbf{\color{#C90}{\phi} \, = \, \frac{\sqrt{5} + 1}{2} } \approx 1.618033989\\ \large \mathbf{\color{#C90}{\varphi} \, = \, \frac{\sqrt{5} - 1}{2} } \approx .618033989\\\]

\[ \large .5 \times 5^.5 + .5\]

It is known by many names. The Divine Proportion The Golden Ratio, Mean, Section

Euclid Extreme and Mean Ratio

A proportion is a relationship between two values. Such as 1:2 or a:b or a/b

**Two values are in The Divine Proportion when the ratio of the lesser value over the greater value is equal to the greater value over the sum of the lesser and greater value.**

In other words, when a segment is sectioned into the Divine Proportion, the parts are in a harmonic relationship to the whole. Setting up a Harmonic Rhythm.

The very nature of the Golden Ratio is harmonic resonance

]]>The Golden ratio is a geometric mean

a geometric mean is a relation between 3 values \(a, b & c\)

in a golden mean, we set the value of \(c = a + b\)

]]>