1st semseter portfoilio 2012/13

Athanasios Korras Ds10

__SyStemS

•Sierpinski Gasket •Sierpinski Carpet

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 ex-amples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.

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 dem-onstrated that this fractal is a uni-versal 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 cor-responding universal curve is the Menger sponge, a higher-dimen-sional 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.

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 cor-ner. 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.

3. Repeat step 2 with each of the smaller triangles

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

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

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

The log spiral is defined in polar coordinates by

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.

•Romanesco Broccoli •Digital Modelling for a Romanesco Broccoli

Romanesco broccoli, or Ro-man cauliflower, is an edible flower of the species Brassica oleracea, and a variant form of cauliflower. Romanesco broccoli resembles a cauli-flower, 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 approxi-mates a natural fractal; each bud is composed of a series of smaller buds, all arranged in yet another logarithmic spiral. This self-similar pattern contin-ues 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 Ro-manesco broccoli form a cone-like shape, so some of the input parameters for the broccoli should relate to this underly-ing structure (image 1).

Image 1

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•SciptforGraSShopperinrhinoceroS

•fractalproGreSSionthrouGhaxonometricS

__Digital RecuRSive geneRation of Koch tetRaheDon expeRiment

• Create the triangle base of the equi-lateral tetrahedron

• Calculate the height of the regu-lar tetrahedron by the function H=sqrt(2/3) x A, where A is the length of the base triangle’s sides. Then, cre-ate the new vertice of the tetrahe-dron.

• Get the 3 vertices of the base trian-gle.

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

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

• Break the data list into 3 segments

• 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

• Create a plane from the mid-points 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. Dis-place the midpoint of the surface both ways. Test for inclusion in the sphere created previ-ously. 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

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

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

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

1Start Shape 2 3 4


cReating papeR component

__initial moDelling appRoacheS

__Koch tetRaheDRon viSualiSation

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1. Fold back along segment AB so the back of vertex E coin-cides with the back of vertex D.

2. Fold back along segment AC so the back of vertex F coin-cides with the back of vertex D.

3. Fold forward along segment GB so vertex E meets vertex A.

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

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

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

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Cutout paper component

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__KoSch tetRaheDRon fouR-face RegeneRation DiagRam

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_DiStoRtion of the BaSic RuleS

1st Recursion 2nd Recursion 3rd Recursion 4th Recursion


•merkabaandthefloweroflife •metatron’Scube

• 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 sur-rounded 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 fig-ure composed of multiple evenly-spaced, 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.

Metatron’s Cube is a three-dimensional geometric figure created from 13 equal circles with lines from the center of each circle extend-ing 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 hu-man experience during the exile from Eden, has led some schol-ars to portray Metatron as the means by which humanity was given knowledge of YHVH; presumably implying that study of Me-tatron’s cube would be necessary to understanding the tree of life.

Construction of all Platonic Solids on Metatron’s Cube

Metatron’s Cube Star Tetrahedron Cube Octahedron Dodecahedron Icosahedron

__cultuRal context

__Koch tetRaheDRon conStRuction appRoach anD viSualiSation

Starting Tetrahedron

1st Recursion

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Starting Tetrahedron

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_“Fractal cult” pipeD conStRuction manual (1 of 2)

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_“Fractal cult” pipeD conStRuction manual (2 of 2)

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_“Fractal cult” timBeR conStRuction manual (1 of 2)

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_“Fractal cult” timBeR conStRuction manual (2 of 2)

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__laSeRcut/cnc componentS foR 1.5 : 1 moDel

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__1.5 : 1 mDf moDel

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__photoRealiStic RepReSentation

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Documents

1st semseter portfoilio 2012/13