Abstract Geometry

Evolving Technical Notes

Elementary Shapes

We’ll be working with several types of objects: points (P), line segments (L), equilateral triangles (T), squares (Sq),  and circles (Cir). A line segment that connects points a and b is denoted as [a, b] and is treated as a singular object. The notation (a,b) will specifically refer to a pair of objects. Our primary relation of interest is connectivity, which we will represent with the letter C.

For instance, a triangle constructed from three interconnected points a, b, and c can be described as:

<a, b, c; P(a), P(b), P(c), C(a, b), C(b, c), C(c, a) >

The triangle’s reflectional and rotational symmetries align with the Abstract Geometry morphisms of the structure into itself. Such a morphism of an object into itself is termed an automorphism.

Drawing from Tarski’s axiomatization of geometry rooted in predicate logic, we’ll introduce the special relation of ‘betweenness,’ denoted as B(a,b,c). This implies that point ‘b’ lies between points ‘a’ and ‘c.’ To define all points lying between ‘a’ and ‘b,’  set notations and pattern variables are used. A set comprising objects a,b,c,d is described as {a,b,c,d}. By using pattern variables like {X} followed by one or more predicates containing variable X, we can characterize a set of all ‘X’ meeting the predicate conditions. This allows us to define a line segment [a,b] as the set of all points ‘X’ situated between ‘a’ and ‘b’:

[a, b] = <{X}, B(a, X, b)>

Here, the set {X} functions as a distinct object. To account for situations where one set is contained within another or one shape is contained within another, we’ll introduce a specialized relation, “in”:

in({a, b}, {a, b, c})

in(a, {a, b, c})

With set notations, we can describe squares and triangles as follows: 

Triangle and square

<a, b, c; T({a, b, c})> 

<b, c, d, e; Sq({b, c, d, e})> 

<{X}; Sq({X})>

To describe circles and centered regular polygons, we’ll regard an object as comprising one object “a” and set {X}. Such an object is portrayed as (a, {X}) or a, {X}. Using this representation, we can discuss a pattern shared by both circles and regular polygons:

abstract geometry centered square

< (a, {X}); R(a, X) >

< a, {X}; R(a, X) >

Depending on the cardinality of set {X}, this pattern could represent either a polygon or a circle. This pattern is what we’ll term an “abstract centroid” with the abstract radius R.

For an object (a, {X}) to be identified as a circle, we’ll need to append another predicate:

< (a, {X}); R(a, X), Cir(a, {X}) >

We have the flexibility to omit the relation R(a, X) and/or introduce details about the circle’s radius, which will be elaborated on later. Comparable notations can be employed for triangles, squares, and regular polygons that designate object a as their central point:

< (a, {X});  Sq(a, {X}) >

< (a, {b, c, d, e});  Sq(a, {b, c, d, e}) >

Using these notations, a square inscribed in a circle can be described as:

< a, {X}, {Y};  Sq(a, {X}), Cir(a, {Y}), in({X, Y}) >

 

Part-Whole Relation and Connectivity

A line segment or a triangle are objects in their own right. We will describe this metamorphosis as follows:

<l; L(l)> = <a, b ; P(a), P(b)  C(a, b) >

< m; T(m)> = <a, b, c ; P(a), P(b), P(c), C(a, b), C(b, c), C(c, a) > 

We may use predicates that mix and match the identity of the whole object and identity of its part. Here are examples:

Cir(c, (a, {X})) -A circle c has center a and circumference points {X}

T(t, (a, {b, c, d})) – A triangle t has center a and points {b, c, d}

Center(a, c), Center(a, t) – a is a center of a circle c and triangle t.

T(t), HasPart(t, a), HasPart(t, b), HasPart(t, c) – a triangle t has parts a, b, c, d.

A triangle can be also described as three connected line segments:

<a, b, c ; L(a), L(b), L(c), C(a, b), C(b, c), C(c, a) > 

This leads to a reflection on the versatile nature of the connectivity relation ‘C’. It’s worth exploring the multiple interpretations of this relation, such as two points linked by a line or two segments intersecting at a point. For instance, connecting four triangles side by side can give rise to a tetrahedron, while linking the edges of eight triangles can form an octahedron. Ambiguity is indeed an intrinsic characteristic of human language.

Lengths, Angles, Ratios, Areas, and Perimeters

In symbolic logic, numeric properties of objects would normally be described in terms of functions. Say the fact that the angle between segments a and b is equal to A would be described as Ang(a,b) = A.

Oftentimes we care about the equality of two angles or ratios on different shapes rather than about the specific values. Moreover, equality should not be precise. If our angles and ratios are predicates we can still think of morphism as mapping that preserves predicates. For that reason, we use non-standard notations and describe angles, lengths, and ratios as labeled relations. The fact that the angle between segments a and b is equal to A would be described as  AngA(a, b).

We will prefix the relation for length with Len, for ratio with Rt, Ar for area and Per for perimeter.   We endow a number with the type of measurement which this number describes. The purpose of this deviation from standard notations of logic is to keep the picture of morphism and automorphism simple. Using such notations we can describe circle of radius 1 and perimeter 2???? as follows: 

< (a, {X}); Cir(a, {X}), Len1(a, X), Per2????((a, {X}))>

Here is the advantage these esoteric notations bring to our framework. Consider a circle c and a square sq with the same area D. We represent area measurement as a predicate ArD. Because the circle and the square have a common predicate ArD they have one partial isomorphism

<c;    Cir(c),  ArD(c) >

   |

<sq;  Sq(sq), ArD(sq) >

This morphism preserves patten

<X;  ArD(X) >

We could have described such morphism with standard logic notations but that would look complicated.

Sacred Geometry and Abstract Geometry

From the standpoint of modern Math, squaring of a circle  is just another geometrical problem. From the standpoint of Sacred Geometry, it is an important mystery.  In Sacred Geometry shapes have meaning. A square may symbolize material form or rational intellect traditionally or left brain intellect. A circle may symbolize soul, spirit or right brain intellect. When such is the meaning of the symbols then squaring a circle may mean harmonizing matter with soul or rational male intellect with female right brain intellect. 

Why are squares and circles with the same area considered to be harmonized? Abstract Geometry is the only framework that can provide a meaningful mathematical answer to this question.  Abstract Geometry elevates the relation between square and circle of the same area to the relation between two squares (or between two circles) because all of these relations are Abstract Geometry partial morphisms. Abstract Geometry describes Harmony implied in Sacred Geometry with the aid of partial morphisms. 

The total Harmony associated with the system is a set of partial morphisms along with their degrees. 

As morphisms have degrees assigned to them, one morphism can create more Harmony than the other. A square and a circle with the same center are more harmonious then ones which have different centers. Consider moving square and circle from previous subsection to a common center a and making a new partial morphism:

<c,   a;  Cir(c),  Center(a, c), ArD(c) >

   |     |

<sq, a; Sq(sq), Center(a, sq), ArD(sq) >

This morphism preserves pattern:

<X, Y; Center(Y, X),  ArD(X) >

This morphism has degree 4 as it maps two objects across 2 predicates. The original morphism had degree 1. So, the latter shape contains more Harmony than the former.

Abstract Geometry reveals important mysteries of Platonic Solids related to their duality.  If we use the same predicate C for different types of connected objects we may discover some fascinating morphism between Platonic Solids and their duals. E.g. an octahedron is dual to a cube. An octahedron described as 8 triangles connected side by side will have a morphism to a cube described as 8 points connected by lines. 

Why do morphisms discovered with the help of Abstract Geometry have some mystical meaning? In the next post on beautyandai.com I’ll argue that Abstract Geometry captures the essence of the primordial forms discussed by Plato. He rightfully compared Forms to shapes drawn by geometers because the second are images of the first. 

Abstract Geometry Examples

Below are a few instances of structures within Abstract Geometry.

Three points a, b, and c, where point b is between points a and c:

<a, b, c; B(a, b, c) >

Golden ratio division of segment [a, c] by point b:

<[a,b], [b,c], [a,c]; B(a, b, c), Rtφ([a,c],[a,b]), Rtφ([a,b],[b,c]) >

An equilateral triangle with a side length of 1:

<a, b, c; C(a, b), C(b, c), C(c, a), Len1(a, b), Len1(b, c), Len1(c, a)>

Two concentric circles with distinct radii R1 and R2, where the first is twice the size of the second:

< (a, {X}), (a, {Y}); LenR1(a, X), LenR2(a, Y), Rt2([a, X], [a, Y])>

A triangle with vertices a, b, c inscribed in a circle centered at m:

<a, b, c, m, {X}; Cir(a, {X}), T(m, {a, b, c}), in({a, b, c}, {X})) >

An abstract triangle inscribed inside an abstract circle with center m and a relation R:

<a, b, c, (m, {X}); Cir(a, {X}), R(m, X), C(a, b), C(b, c), C(c, a), R(m, a), R(m, b), R(m, c) >

By replacing the relation R with Len1, this structure can be transformed into a circle of radius 1 with an inscribed triangle.

It’s vital to emphasize that Abstract Geometry comprises structures tailored to describe images, specifically focusing on their geometric properties. Similar to structures that depict situations and imagery, Abstract Geometry’s structures should remain adaptable. Predicates can be added, removed, or altered, and one structure can be converted to another using transformation rules. If Abstract Geometry has any theorems, they would manifest as transformation rules.

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