#self-similar

[cadmar]

As Above, So Below

Scientists, using power spectrum comparison and  estimating the complexity of a network by measuring how difficult it is to predict its behaviour,  have determined that billions of galaxies connections and the trillions of neurons in the brain are self-similar.  The only difference is in scale.

Symbols are tools that can connect these two realms!

http://nautil.us/issue/50/emergence/the-strange-similarity-of-neuron-and-galaxy-networks

[sachifoundation]

Relationship between the effort and Results depends on the measures of the cities of Knowledge and Loving kindness

The relationship between the effort and result depends on the measures being used in the self-similar cities of Knowledge and Loving-kindness. Within the city of knowledge; every line of effort must have its results. This is because of the various discourse vehicles that facilitate response-receptivity to the elements of the collective action also carry the practical knowledge for sustainable solutions within assigned time space. We provide grants support for these types of projects subjects to our own limits of financial and organisational capacity. We consider efforts without results as the line-up of the discourse vehicles within the city of loving kindness. For us, they are not yet loaded with sufficient knowledge that marshals all the necessary impact evaluation grant processes as it was required by the measures in the city of knowledge. For these unaccomplished projects, we offer encouragements for the conviction and contributing toward the overall group discourses development.

[birdthatlookslikeastick]

Here are two pictures of what this post looked like as a draft.

[sachifoundation]

New Economic order and Self-similar evolution of the Civilizations

At the heart of the emerging new world order is a global network of clusters that supports new ways of working, learning and sharing. It is a global cultural network of the self-similar evolution of the civilization natural order where each culture becomes both a discourse and an active node in delivering happiness for all. It feeds on the Ridvan festival self-similar energy supply network to sustain the large-scale coherence between all its nodes and their nodal growth. Building on this manner, we can easily visualize the structural type of self-similarity in the rise of new economic order for the new way of living and generating wealth. This emerging economy is built on self-similarity patterns associated with “work as a worship” and “unity of craftsman efforts” to produce goods and services.

[curiosamathematica]

Penrose tilings are non-periodic tilings, generated by an aperiodic set of prototiles, implying that a shifted copy of a Penrose tiling will never match the original. They are named after Roger Penrose, the mathematician and physicist who investigated these sets in the 1970s.

These tilings have some remarkable properties, apart from aesthetically pleasing:

Non-periodicity, meaning it lacks any translational symmetry.

Self-similarity, meaning the same patterns occur at larger and larger scales.

In the top image, you see Roger Penrose himself in the foyer of the Mitchell Institute Building at Texas A&M University, standing on a Penrose tiling.

The main plaza of the Finnish Heureka science museum is paved with a Penrose tiling of kites and darts.

If you're interested in the question why this 5-fold rotational symmetrical patterns are aperiodic, you should check out the crystallographic restriction theorem: for every discrete isometry group in two- and three-dimensional space (mapping every point to a discrete subset, i.e. a set of isolated points) which includes translations spanning the whole space, all isometries of finite order are of order 1, 2, 3, 4 or 6. The finite orders in higher dimensions are restricted as well: in four- and five-dimensional space, one can only find finite orders 1, 2, 3, 4, 5, 6, 8, 10 and 12. In six and seven dimensions, possible finite orders are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 or 30.

[mucholderthen]

DO IT YOURSELF FRACTAL  How to Make an Equilateral Triangle into a Sierpinski Carpet As described in the Wikipedia (see also here and here)

The Sierpinski triangle – named after the Polish mathematician Wacław Sierpiński (who described it in 1915) – is one of the basic examples of self-similar sets (and, so, of fractals).

A self-similar set is a mathematically generated pattern that can be reproduced at any magnification or reduction.  Put another way, it looks similar (or the same) no matter how near or far you are.

A Sierpinski "Carpet" is the same, except you can use any shape.

[1] Start with any triangle in a plane. 

The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image, bottom left).

[2] Shrink the triangle to ½ height and ½ width.

[3] Make three copies of the shrunken triangle.

[4] Position the three shrunken triangles so that each triangle touches the two other triangles at a corner (second from left).     

By doing this, a central hole emerges. This is because the three shrunken triangles can between them cover only 3/4 of the area of the original.

[5] Repeat steps 2, 3, and 4 with each of the smaller triangles. Repeat this step [step 5] as long as desired (as shown in the three images on the bottom, center to right).

IMAGES TOP Wikipedia: equilateral triangle dividing into three copies at smaller and smaller scales to create a Sierpinski carpet   BOTTOM  Wikipedia:  constructing the Sierpinski triangle

IF YOU WANT TO RE-BLOG ONLY THE GRAPHICS, FINE - just leave the image credits at the bottom if possible.

[jessehensel-blog]

Today I was finally able to render part of the series in Autodesk. This is the 4th iteration of the pyramid aspect of the series. I hope to crank out the 3rd iteration of both the pyramid and the tetrahedron tomorrow.

[brandistrickland]

Brandi Strickland Self-Similar, 2009 Collage, acrylic, & gouache on board, 12×12" Print // Inner Space Series