1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987 et cetera. Notice anything special about this series? It is additive. 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21. This is the Fibonacci series. Each Fibonacci number is the sum of the two Fibonacci numbers preceding it.
What makes the Fibonacci series even more interesting is its direct relation to the divine proportion. When a Fibonacci number is divided by the number preceding it in the series, the ratio gets closer and closer to "phi" (1.6180339887498948482045...). For example, 144/89=1.617977, 233/144=1.618055.
The larger the two consecutive Fibonacci numbers, the closer the ratio gets to the divine proportion. Phi is an infinitely long number, and can be expressed as ((1 + √5)/2).
Phi also has interesting properties: phi² = 2.61803398874989.., and 1/phi = 0.61803398874989..
When x²-x-1=0, x=1.6180339887498948482045...
How does this relate to nature? Look at the bottom of a pine cone, the head of a daisy, or the head of a sunflower. Notice that there will be clockwise spirals and counter-clockwise spirals. Each seed is a member of both spirals. A very large majority of the time, the number of clockwise spirals and counter-clockwise spirals will be two consecutive Fibonacci numbers.
The number of petals on a flower is often a Fibonacci number. For example, a sunflower or daisy often have 21, 34, or 55 petals. Vinca, larkspur, and columbine have 5 petals. Coreopsis, bloodroot, cosmos, and delphinium have 8 petals. Marigold and ragwort have 13 petals. Chicory and aster have 21 petals. Plantain and pyrethrum have 34 petals.
Self Tessellation
Self tessellation means placing a shape side by side without leaving any gaps. Only 3 polygons can self tessellate. They are the triangle, quadrilateral, and hexagon. When 6 circles of the same size are placed perfectly around a center circle (altogether 7 circles), a perfect hexagonal pattern emerges. When honeybees are building their comb, they make circular cells that, when placed side by side, will naturally form into a perfect hexagon. Why do the honeycomb cells form hexagons instead of triangles or squares? Why do they always form a 6-sided polygon?
The answer is quite simple. Forming hexagons instead of any other shape is the most efficient thing possible when it comes to using resources.
Here is a mathematical example: there are 3 polygons (a triangle, a square, and a hexagon). All 3 shapes have the same perimeter of 24 units. That means that the triangles area would be 27.713 units², the squares area would be 36 units², and the hexagons area would be 41.569 units². Using the same "materials", the hexagon will enclose a bigger area. It is naturally more efficient than the other 2 self tessellating polygons.
The Pentagon
A regular pentagon (a 5 sided polygon with all interior angles at 108 degrees) has more to it than meets the eye. Remember the divine proportion? (1.61803398874989 : 1) The pentagon has that proportion embedded in it.
If you take any vertex in the pentagon and connect it to two other vertices (create diagonals), you will get an isosceles triangle. This triangle will have a base of 1 unit, and the other two longer sides will be exactly 1.6180339887498948482045 units.
More simply put, if you connect any vertex to another vertex in a pentagon, the length of that line will be "phi", 1.6180339887498948482045 units, given that each side length of that pentagon is 1 unit.
Interestingly enough there is a connection to the pentagon and growth. To explain, we must look at a simple method of creating a golden rectangle (a rectangle with the short side length as 1, and the longer side length as 1.6180339887..) with Fibonacci numbers. Start with a perfect square with side lengths 1; put another identical square right next to it; then put a square with side lengths 2 right on top of the other 2 squares; then put a square with side lengths 3 right next to the other 3 perfectly fitting squares. Continue this process with successive Fibonacci numbers (1,1,2,3,5,8,13,21,34..), and the more Fibonacci numbers used, the closer the side lengths of the rectangle being created get to 1 and 1.6180339887.
Take that golden rectangle (which was created with a growth of squares fitting most efficiently), and flip it so the side length of 1 is the base. Take the top side, and get rid of it. Now you have an open shape that has two 90 degree angles, base length 1, and the other two side lengths being 1.6180339887. Take the two longer sides and bring them together at the same rate to form a triangle. That triangle is the same exact triangle that is made when connecting vertices in a regular pentagon.
The Platonic Solids
There are 5 Platonic solids; the tetrahedron (4 equilateral triangles), hexahedron (6 equilateral squares), octahedron (8 equilateral triangles), dodecahedron (12 equilateral pentagons), & icosahedron (20 equilateral triangles). The 5 Platonic solids' faces are congruent regular polygons, with the same number of faces meeting at each vertex (every corner of the shape looking the exact same). These 5 three-dimensional shapes are very important in sacred geometry in many ways.
The dodecahedron can be created using 3 interlocked rectangles. All 3 rectangles would have congruent 90 degree angles, shorter side with length 1, and the longer side with length 2.6180339887498948482045... Each rectangle would essentially epitomize one of the 3 spatial dimensions (one rectangle going up/down, another going left/right, and the last going forward/back). This structure of 3 interlocked rectangles will fit perfectly inside a dodecahedron.
Using the same process but replacing side length 2.6180339887 with 1.6180339887 will create a structure that will fit perfectly inside an icosahedron. Curiously, 1.6180339887 is the square root of 2.6180339887, and the icosahedron is the dual of the dodecahedron, and vice versa.
The corners and faces of the Platonic solids are another area of interest. All five of them have a connection to the equilateral triangle: the tetrahedron is made up of equ. triangles and its corners are all equ. triangles; the hexahedrons corners are equ. triangles; the octahedron is made up of equ. triangles; the dodecahedrons corners are equ. triangles; and the icosahedron is made up of equ. triangles.
In 3 dimensions there are 5 regular shapes made up of equilateral triangles: the Platonic solids. In 2 dimensions there are only 2 regular shapes made up of equilateral triangles: the equ. triangle itself and the hexagon. 2 dimensional projections of the Platonic solids will be either an equ. triangle or an equ. hexagon depending on which Platonic solid it is. If it is the tetrahedron, its shadow can be an equ. triangle; if it is the hexahedron, octahedron, or icosahedron, then the shadow can be an equ. hexagon; the dodecahedrons shadow, however, cannot be an equ. hexagon, but can fit perfectly inside one.
Connecting the midpoints of each face of a Platonic solid will give you its "dual". Basically, consider the center of each face/polygon as a point, then connect those points. The dual of the tetrahedron is itself, the dual of the hexahedron is the octahedron, the dual of the octahedron is the hexahedron, the dual of the dodecahedron is the icosahedron, and the dual of the icosahedron is the dodecahedron. There is another way to look at the dual relationships between the Platonic solids. The tetrahedron is made up of equ. triangles, and the corners are equ. triangles. Switching the shapes of the faces and the shapes of the corners will give you the shapes dual. This is why the tetrahedron is the dual of itself. The hexahedron is made up of squares, and the corners are equ. triangles; the octahedron is made up of equ. triangles, and the corners are squares. The dodecahedron is made up of pentagons and the corners are equ. triangles; the icosahedron is made up of equ. triangles, and the corners are pentagons.
The Divine Proportion & Equilateral Triangle
At first glance one would not think to find the divine proportion in the equilateral triangle; certain ratios such as 1/2 or 1/√3 already exist in this shape. Nonetheless, the divine proportion is there.
If you inscribe an equilateral triangle in a circle and draw lines that divide that triangle into fourths, phi emerges. Curiously, this directly connects the tetrahedron to the divine proportion, being that the equ. triangle inscribed in the circle below is also a map of an unfolded tetrahedron.
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