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__Systems • Sierpinski Gasket The Sierpinski Gasket, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.

• Sierpinski Carpet The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions. Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional plane, is homeomorphic to a subset of the Sierpinski carpet. For curves that cannot be drawn on a 2D surface without self-intersections, the corresponding universal curve is the Menger sponge, a higher-dimensional generalization.

1. Start with any triangle in a plane. The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis.

1. The construction of the Sierpinski carpet begins with a square.

2. Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner. Note the emergence of the central hole - because the three shrunken triangles can between them cover only 3/4 of the area of the original.

2. The square is cut into 9 congruent subsquares in a 3-by-3 grid, and the central subsquare is removed.

3. Repeat step 2 with each of the smaller triangles

3. The same procedure is then applied recursively to the remaining 8 subsquares, an infinitum.

• Romanesco Broccoli

• Digital Modelling for a Romanesco Broccoli

Romanesco broccoli, or Roman cauliflower, is an edible flower of the species Brassica oleracea, and a variant form of cauliflower. Romanesco broccoli resembles a cauliflower, but is of a light green colour and the inflorescence (the bud) has an approximate self-similar character, with the branched meristems making a logarithmic spiral. In this sense the broccoli’s shape approximates a natural fractal; each bud is composed of a series of smaller buds, all arranged in yet another logarithmic spiral. This self-similar pattern continues at several smaller levels.

Romanesco Broccoli has great complexity and depends upon mathematical concepts like the logarithmic spiral. The flower is composed of spirally arranged segments which are identical copies of the whole flower. The copying process continues infinitely as a 3D fractal form. The buds of our Romanesco broccoli form a cone-like shape, so some of the input parameters for the broccoli should relate to this underlying structure (image 1).

level 1

level 2

Hc Image 3

The log spiral is defined in polar coordinates by

Rc Image 1

Image 2

with e being the base of natural logarithms, and a and b being arbitrary positive real constants difined for the spiral (image 2). First, you have to create the desired 2D spiral before moving to 3D. This 2D spiral was based off image 3.

Athanasios Korras Ds10


__Digital Recursive Generation of Koch Tetrahedon Experiment • Scipt for Grasshopper in Rhinoceros • Create the triangle base of the equilateral tetrahedron

• Join all the input Breps and create the bounding box of them. Take the volumetric midpoint and create a sphere originating from this point with a radius that is defined by another process

• Break the data list into 3 segments

• Calculate the height of the regular tetrahedron though the formula sqrt(2/3)*x, where x is the length of the tetrahedron’s side.

• Get the 3 vertices of the base triangle.

• Calculate the height of the regular tetrahedron by the function H=sqrt(2/3) x A, where A is the length of the base triangle’s sides. Then, create the new vertice of the tetrahedron.

• Calculate the number of new surfaces that will be created depending on the number of recursions I want with the formula (3^(x+1)), where x is the number of recursions wanted. Then I compare the number to the number of surfaces that we are getting. If it is greater then the loop continues.If it is equal then it stops.

• Create the new equilateral triangles that will form the sides of the regular tetrahedron by connecting the vertices we got in steps 2,3.

• Collect all the surfaces together and flatten their data tree so we can feed our loup with clean starting data for the next recursion.

• Explode the Breps to get their edges and then get the midpoints of those edges.

• Create a plane from the midpoints we just got and reposition its origin to be the midpoint of the polyline created through these midpoints. Decompose this plane in order to get its normal.

• Amplicate the normal by the height of the new regular’s tetrahedron, found earlier. Displace the midpoint of the surface both ways. Test for inclusion in the sphere created previously. Get the one that has a false value, thus being outside of the geometry.This is the new vertice of our tetrahedron.

• Create the surfaces between the initial edges and the new vertice in order to get the new regular tetrahedron

• Fractal Progression through Axonometrics

Start Shape

1

2

3

4

Athanasios Korras Ds10


__Koch Tetrahedron Visualisation

E

D

D

F

B

E

D

G

A

A

E

B

C

D

D

D

F

E

F

__Initial Modelling Approaches Creating Paper Component E

1. Fold back along segment AB so the back of vertex E coincides with the back of vertex D.

E

D

D

F

B

B

B

C F

A

B

C

B

C C

B

A

A

G

C

E

E

G

A

B

C

A

A

D

E

F

E

B

A

A F

C

E

5. Fold forward along segment BC so vertex D meets vertex A.

E

G

A

1

4. Fold forward along segment HC so vertex F meets vertex A.

6. Fold the creased template so that edge AG coincides with edge AH; G meeting H. See it here.

D

D

D

2. Fold back along segment AC so the back of vertex F coincides with the back of vertex D. 3. Fold forward along segment GB so vertex E meets vertex A.

B

B

A

A

A

G

A

D

D

A

5

C

B

C

C

C

B D

C

D

Cutout paper component C

B A C

2

C

G

D

A

A

H

A

A

H

A

A

A

E

F

H

A

G

H

G

6

A

D

F H

A

A

C

E

E

E

F

4 D

F

EB

F

F G

H

D

C

B C

A D

F A

H

A

E

F G

A

A

D

A

D

D

F

H

B

A

3

B

G

H

E

H

Athanasios Korras Ds10

H

F

F G

H


__Kosch Tetrahedron Four-Face regeneration Diagram

1

2

3

4

_Distortion of the Basic Rules

1st Recursion

2nd Recursion

3rd Recursion

4th Recursion

Athanasios Korras Ds10


__Koch Tetrahedron Construction Approach and Visualisation

Starting Tetrahedron 1st Recursion

Starting Tetrahedron 1st Recursion

2nd Recursion

2nd Recursion

3rd Recursion

3rd Recursion Connections

Connections

__Cultural Context • Merkaba and the Flower of Life

• Metatron’s Cube

Metatron’s Cube is a three-dimensional geometric figure created from 13 equal circles with lines from the center of each circle extending out to the centers of the other 12 circles. This image creates the Fruit of Life. Six circles are placed in a hexagonal pattern around a central circle, with six more extending out along the same radial lines. Metatron’s Cube is a figure in sacred geometry. Its name makes reference to Metatron, an angel mentioned in apocryphal texts. These texts rank Metatron second only to YHWH in the hierarchy of spiritual beings. The derivation of Metatron’s cube from the tree of life, which the Talmud clearly states was excluded from human experience during the exile from Eden, has led some scholars to portray Metatron as the means by which humanity was given knowledge of YHVH; presumably implying that study of Metatron’s cube would be necessary to understanding the tree of life.

• Merkabah, also spelled Merkaba, is the divine light vehicle allegedly used by ascended masters to connect with and reach those in tune with the higher realms. “Mer” means Light. “Ka” means Spirit. “Ba” means Body. Mer-Ka-Ba means the spirit/body surrounded by counter-rotating fields of light, (wheels within wheels), spirals of energy as in DNA, which transports spirit/body from one dimension to another.

• The Flower of Life is the modern name given to a geometrical figure composed of multiple evenlyspaced, overlapping circles. They are arranged to form a flower-like pattern with a sixfold symmetry, similar to a hexagon. The center of each circle is on the circumference of six surrounding circles of the same diameter. It is considered by some to be a symbol of sacred geometry, said to contain ancient, spiritual value depicting the fundamental forms of space and time.

Construction of all Platonic Solids on Metatron’s Cube

Metatron’s Cube

Star Tetrahedron

Cube

Octahedron

Dodecahedron

Athanasios Korras Ds10

Icosahedron


_“Fractal cult” Piped Construction Manual (1 of 2)

• Components

• Construction Steps 3-way Joint

x56

5-way Joint

5

2

3

5

6

7

9

10

11

x48

8-way Joint

x54

Pipes

x12

x48

x360

Athanasios Korras Ds10


_“Fractal cult” Piped Construction Manual (2 of 2)

4

8

12

13

Athanasios Korras Ds10


_“Fractal cult” Timber Construction Manual (1 of 2)

• Construction Steps

• Components Faces and Joints in 2d

1

2

4

5

7

8

Joints

x24

x195

x171

Faces

x9

x27

x162

Athanasios Korras Ds10


_“Fractal cult” Timber Construction Manual (2 of 2)

3

6

9

10

Athanasios Korras Ds10


__Lasercut/CNC Components For 1.5 : 1 Model

Materials • 1220 x 2440 mm MDF BOARD x 3 • 1220 x 900 mm MDF BOARD x 10

x4

x3

x2

x1

x1

x2

Athanasios Korras Ds10


__1.5 : 1 MDF Model

Process

Athanasios Korras Ds10


__Photorealistic representation

Day

NIght

Plan View

Athanasios Korras Ds10


__Photorealistic representation

Day

NIght

Athanasios Korras Ds10


__Photorealistic representation

Day

NIght

Athanasios Korras Ds10


__Photorealistic representation

Day

NIght

Athanasios Korras Ds10

1st semseter portfoilio 2012/13  

1st semester portfolio in University's of Westminster DS10, 1012/13

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