Geometry, Nature & Architecture

All of nature evolves out of simple geometric patterns incorporated within the molecular "seed" structure. Each of these basic patterns contains information that enables animals, plants, minerals (and humans) to develop into complex and beautiful forms, each with an intrinsic awareness of its location in space and time. Being part of nature, we have a relationship with it at the cellular level which is experienced vibrationally, and which is nurturing. When these seed patterns are incorporated into our architecture, a vibrational exchange takes place between the building and its occupants in a way that is similar to the connection we have with nature, and which leads to a sense of well being.

Can you find Geometric Shapes in Buildings and Structures? 

In this example, six basic elements of geometry (point, segment, ray, line, angle, and triangle) have been identified within the bridge.

How many three-dimensional shapes (cube, tetrahedron, cylinder, etc.) can you find in this computer graphic of the Parthenon?

Can you find the "Golden Sections" within?

The Golden Section

The Golden Section refers to rectangles that have a ratio of 1:1.6. This means that if the width of a rectangle is 1 foot long, the length will be 1.6 feet long. (The number 1.6 is sometimes referred to as "phi", and it looks like the letter o with a line through it (ΓΈ).

The Golden Section (or Golden Ratio or Golden Mean or Golden Rectangle) appears in a lot of ancient Greek architecture and has been analyzed extensively by the famous mathematician Fibonacci. The ratio of 1:1.6 is said to be "pleasing to the eye."

Geometry and Nature

In the world of natural phenomena, it is the underlying patterns of geometric form, proportion and associated wave frequencies that give rise to all perceptions and identifications. Therein lies our fundamental capacity to relate, to interpret and to know. Most of the interpretations are of a graphic nature.

The Nautilus Shell Spiral

The shell of the chambered nautilus is a symbol of beauty and proportional perfection. In this exercise you will experience the essence and energy of the spiral displayed in the unfolding of the nautilus shell. 

The proportion of this nautilus shell is consistent through all of the relationships of the shell, so once you discover the secret ratio, you will see clearly why this particular gem of nature is such a treasure. The proportion referred to is often called the "Golden Proportion" or the "Golden Ratio" and can be mathematically approximated by the ratio of 1:1.618. 

The nautilus shell is also called the "Divine Proportion" and can be found throughout the human body and nature. The drawing of the inner ear and the photo image of a star cluster nebula shown to the right are just twoexamples which can be seen with a microscope or a telescope. There are many examples which can be seen without magnification. 

 

 

  • This beautiful spiral can be easily drawn on a piece of grid paper with a pencil. 

Step 1 - Begin with a small single square, one unit per side, near the lower right-hand corner of the page. Continue adding larger and larger squares in a counterclockwise direction using the following number sequence: 1, 1, 2, 3, 5, 8, 13... as illustrated in the example. This number sequence is often referred to as the "Fibonacci series."

Step 2 - Draw diagonals through every square in a counterclockwise direction. Use the diagonals as guides and draw a smooth spiraling curve from the smallest, one-unit square, outward through your largest square. The proportional relationship of the squares quickly begins to approach the Golden Proportion of 1:1.618...

 

 

If you have a drawing compass, you can set the point at the corner of each square, opposite the diagonal, and construct a beautiful spiraling curve out from the smallest square. The center is known as the "eye of the spiral." 

The Sunflower Mandala: A Universal Growth Pattern of Expansion and Contraction 

The Sunflower Mandala exercise will remind you of a dynamic motion found in the Natural World. Experience the beautiful principle of enfolding and unfolding as you work and participate in a creative process which is timeless and universal. 

  • Method of  Construction

Step 1 -To begin, start with a point on your page, and it draw a vertical   line and a horizontal line perpendicular to it, also through your center point. Next, using your compass, draw a circle with a 3.4-inch radius. Then with the same compass radius, further subdivide the circumference of your circle into 12 segments. Remember... the radius of a circle exactly cuts the circumference into 6 equal segments. It is easy to then divide these arcs in half, resulting in 12 segments. 

Step 2 - Draw light guidelines with your straightedge from the circumference into the center. The corner point of the largest square (with 1-inch side length) should be placed so that it just touches the guideline circle. The rest of the exercise will unfold to you as you work. 

Step 3 - Use your compass to draw more circular guidelines as you need them and as you progress proportionally in toward the center. If your guidelines are drawn lightly with a pencil they will not distract from the completed image. 

This type of pattern is referred to as a Mandala, a Star Polygon, a Rose Window, and other names as well, depending on the culture, tradition or application where it is used. The image can be developed into a Tessellation or Field Pattern if repeated 

The Snowflake Pattern

Snowflake Pattern illustrates a beautiful pattern known throughout the world. It can be found in every culture on every continent.

 

 

 

 

 

The seed pattern for all snowflakes is based on the regular polygon known as the hexagon. The hexagon is a unique shape with six equal sides and six equal interior angles. 

If you allow the symmetry of the six-fold geometry to be your guide, virtually any image you create will show forth the essential nature of the snowflake. See what you can do. There are many other examples in the nature kingdom where you can observe this hexagonal geometry.

Six-Around-One

One of the most interesting and useful patterns in Nature is commonly referred to as "Six-Around-One." It is known throughout the world and often referred to by other names, such as "Closest Packing of Circles," "Hexagonal Packaging," and "Tessellating Hexagons." The pattern is very common.

The arrangement of the 7 equal circles clearly shows the relationships which occur in a Six-Around-One pattern. It is one of the most efficient and stable systems employed by nature in two-dimensions as well as three-dimensions in an omni-directional universe.

 

 

 

 

The ancient Greek mathematicians and philosophers were well aware of this pattern and the principles it displayed.They observed it in nature and taught it in their schools. One of their methods for demonstrating this pattern was to have the students manipulate round pebbles or glass beads in a bowl of sand. Another way to experience it is with 7 coins of equal size, or by drawing it with a compass.

 

 

 

 

 

 

Geometry in Architecture

Our ability to create music and interpret sounds is initially a mechanical interpretation of wave patterns, not through quantitative differences but proportional differences. 

The architecture of the buildings we inhabit as our dwellings, our work places, our healing and education centres, and our churches and temples, has similar impact on our senses as the architecture of nature.

In understanding how this connection between nature and man made objects works, one should considered the meaning of numbers and their associated geometric forms.

One Begets Two - The Essence Of Duality

If we look at the geometry the circle represents God, the circumference being that which encompasses all that is, and the area being all that is. All geometric forms that evolve out of prime numbers are contained within the circle, for it is the circle that defines their symmetry. Implicit within a circle, and its most fundamental product or creation is its diameter. The diameter is a straight line - it defines two points (at its ends, which lie on the circumference), thus symbolising duality or polar opposites. One, in the sense of absolute exclusivity as distinct from one as a unit amongst many, is incomprehensible, and only God is absolute unity. Try to imagine that you alone existed within aninfinite void - you would be incapable of identifying even yourself as there would be no datum point for recognition to occur. One therefore is meaningless, as only God is absolute unity, and one represents creation unmanifested.

Three Is The Power Behind Creation

The next step in creation, moving from 2 points on the circumference of a circle is 3 points. When these are in symmetry with each other, they define the equilateral triangle. Three linear structural elements connected together at their end points form a rigid frame - a fundamental engineering principle. The triad or trinity in esoteric terms expresses a rigid structure, and demonstrates the stability within creation by virtue of the principles of the Father, the Son and the Holy Spirit. All subsequent steps in creation contain triangles.The 3-dimensional version of the triangle is the tetrahedron, a solid object contained within a sphere, but one which is pure of form because it contains 4 faces all of which express the trinity. It has no diagonals,no shortcuts between its defining points. Although it appears to be a 3-dimensional object, it is really multi-dimensional, and represents consciousness on higher planes of reality wherein all manifestations are aware of their existence within the trinity. It also demonstrates how every single addition (point) in creation gives rise to another triangle as part of a larger rigid framework, the basis of construction of a stable universe.

Four Is Manifestation Into Matter
 
Next we have four, the square, symbolizing creation in matter. The square defines the four directions of north, south, east and west, and in its 3-dimensional form, the cube (contained within a sphere) defines the 3 directions of 3-dimensional reality. The square and the cube represent physical manifestation unevolved. The potential that lies within them is implicit. For example, the square's diagonal implies another square, which in turn contains a diagonal with an even larger square. Continued on, this creates a spiral which is the dynamic energy within all creation, and symbolizes involution and evolution, progress towards the infinite, both inwards and outwards. Four also represents the first product of the procreative process that is of multiplication 2 x 2 = 4. 
 
Involution - Five Is The Potential Contained Within Matter
 
The next step with five, brings consciousness into matter by virtue off, the golden proportion. Five points equally spaced on the circumference of a circle give rise to the pentagon, within which lies the pentagram or 5-pointed star. This is generally accepted as representing man - the head, two legs astride, and two arms outstretched. 
The diagonal of a pentagon, in proportional relationship with its side, defines f. This is the purest form of proportion, and thus represents human consciousness in its greatest and most harmonious state. As such, it cannot be defined in absolute terms. Perfect consciousness in matter is something to be sought. An evolved consciousness recognises perfect harmony, but in the imperfect state it is given no status. All creative expressions that emanate from man can be evaluated to the degree in which they incorporate the golden proportion 
 
Evolution - Six Harnesses The Power Of  Spirit And Brings It Into Matter
 
Whereas 5 expresses the potential for harmony hidden within the world of matter is explicit in its representation of symmetry and balance. The hexagon has all its diagonals passing through the centre of the circle and equal to the diameter. Within it lies the hexagram (6 pointed star) and its 3-dimensional form the star tetrahedron. The hexagram is made up of two superimposed equilateral triangles, one inverted. The trinity has replicated itself in the world of matter as represented by the inverted triangle. The hexagram thus presents the challenge for mankind to recognize his own divinity.
 
 
A similar relationship between God and His creations in matter can be seen in the Vesica Piscis, where two circles overlap. A connection exists in these two symbols because the Vesica is the geometric basis for establishing the hexagram. The Vesica can also be seen as the two aspects of God within the material level of creation - masculine and feminine, the polar opposites in humanity arising out of duality. Two circles apart define separation between male and female, but in the overlapping they are integrated. The Vesica thus expresses the powerful potential in mankind to achieve harmony through the blending of the masculine and feminine qualities. The overlap area has traditionally been seen as a fish or the Piscean Christ. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The Link Building
Plane and Elevation
 
 
 
Buildings can have a profound influence on our health and our psychic and spiritual state of be-ing, Harmony and balance, light and colour, relationship to landscape, ecological sympathy, energy efficiency and geometric form are contributing elements of shelter which aspire to be nurturing rather than draining. We resonate at both cellular and consciousness levels with our environment. By creating an environment around us that is supportive to both our inner and our outer senses, we can enhance rather than alienate our human links with nature. Architecture, when employed as a means of embodying principles of universal harmony can sustain us rather than drain us, so that our homes become our havens, and our work places support our creativity. 
 
When architecture incorporates these symbols, they need only be implicit as a theme out of which the design evolves. The power within the symbols is vibrational, and the way they are used should optimally be compatible with the inhabitants. This is the role of the designer, to identify that compatibility. When vibrational symbols are used as a rigid format springing from the mind, their power is limited. The mind is a tool. Only when thought is coupled with intent arising from the heart, does manifestation occur.
This presentation was created by: 
Ankit Bansal

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