H-Tree Fractal - Real Number Line - VLSI

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Bruce Schuman

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Oct 1, 2016, 5:15:12 PM10/1/16
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Speaking of Holy Grail – I recently came across a mathematical idea I find fascinating – for a variety of reasons –

 

Just thought I would see if anybody in ontolog has considered anything like this.

 

When I first saw this “H-tree fractal”, my first interpretation was to see this form as a fractal cascade of boundary values defined in the unit interval – or as a recursive/self-similar cascade of boundary values, where each “T” in the H form defines  a boundary.  This thing can be “open at the top” – so as to “be open to the infinite”, if that is interesting – and also at the bottom (“the infinitesimal”) – both perhaps defined as “unbounded”.

 

https://en.wikipedia.org/wiki/H_tree

 

In fractal geometry, the H tree is a fractal tree structure constructed from perpendicular line segments, each smaller by a factor of the square root of 2 from the next larger adjacent segment. It is so called because its repeating pattern resembles the letter "H". It has Hausdorff dimension 2, and comes arbitrarily close to every point in a rectangle. Its applications include VLSI design and microwave engineering.

 

Every one of these recursive units takes the form of an “interval” with “boundary value cuts” at either end of the T (branches of the H).  The value of the interval can be the range from 0 to 1, and the center-point of the interval can be the 0 point (x,y origin) for the next lowest level.  Keep pushing this form, it’s going to converge to a limit --  of something like continuity, or maybe the “real number line”.  It’s a “binary” form, like a Cantor Set – with a mid-point at each level (the Cantor set is defined by removing the middle third of a recursive series – and that middle third could be defined as a center-point in a digital or 2-state logic).

 

I like this thing as a general-purpose top-down/bottom-up analytical cascade, since it points towards any sort of hierarchical part/whole model.  I like the idea of defining its elements as “cuts” – as per my own intuitive ontological flash (my version of “turtles all the way down”), which I see as the essence of taxonomy: “a cut on a cut on a cut on a cut on a cut…”   I’d say the implications for universal ontology jump up a notch when we consider the concept of cut as a “figure/ground” definition.   Can this structure be closed on itself, as a sealed topological form – maybe like Douglas Hofstadter’s “strange loops”?

 

This principle has significant applications in network and VLSI computer chip design.  It’s “linearly recursive and self-similar” – more or less ultimately simple.

 

Can all abstraction of any sort be defined by this kind of linear recursion?  I am inclined to say yes.

 

There’s a good technical article from the Royal Society:

 

“Scaling theory for information networks” -- http://rsif.royalsocietypublishing.org/content/5/29/1469

 

This thing feels very universal to me.  It’s “holistic”.  Am I seeing a definition of “relative” and “absolute” here?  Can these T intersections be interpreted as nested cartesian coordinate frames in x and y?

 

It’s 100% symmetrical, linear, and self-similar, and every dimension of its form is a unit interval with a center-point.  Is all of that in one integrated format as interesting as I keep thinking it is?

 

I like this analysis because it is tightly mapped to a hard linear grid, and every “cut” defined on it has a “width” or “thickness”  - shown below as w(o) – which I think points to real-world facts of life in symbolic representation (i.e., “constructivist” definitions).  So, I think we’re talking about “actually building one of these”, rather than imagining some abstract logic in our mind and calling that a proof of something.

 

 

“The H-tree: a 2-dimensional hierarchical self-similar branching network that distributes clock signals on an integrated circuit”

 

http://origin.org/onedb/graphics/104669.jpg

 

 

In VLSI design, the H tree may be used as the layout for a complete binary tree using a total area that is proportional to the number of nodes of the tree.[3] Additionally, the H tree forms a space efficient layout for trees in graph drawing,[4] and as part of a construction of a point set for which the sum of squared edge lengths of the traveling salesman tour is large.[5]

 

It is commonly used as a clock distribution network for routing timing signals to all parts of a chip with equal propagation delays to each part,[6] and has also been used as an interconnection network for VLSI multiprocessors.[7] For the same reason, the H tree is used in arrays of microstrip antennas in order to get the radio signal to every individual microstrip antenna with equal propagation delay.

 

The planar H tree can be generalized to the three-dimensional structure via adding line segments on the direction perpendicular to the H tree plane.[8] The resultant three-dimensional H tree has Hausdorff dimension equal to 3. The planar H tree and its three-dimensional version have been found to constitute artificial electromagnetic atoms in photonic crystals and metamaterials and might have potential applications in microwave engineering.[8]

 

 

https://en.wikipedia.org/wiki/Cantor_set

 

https://en.wikipedia.org/wiki/H_tree

 

http://mathworld.wolfram.com/MandelbrotTree.html

 

https://en.wikipedia.org/wiki/Figure%E2%80%93ground_(perception)

 

Buce Schuman

www.origin.org/book.cfm

 

 

From: ontolo...@googlegroups.com [mailto:ontolo...@googlegroups.com] On Behalf Of Gregg Reynolds
Sent: Wednesday, September 28, 2016 1:21 PM
To: ontolo...@googlegroups.com
Subject: -- SPAM --[ontolog-forum] is "eating" universal?

 

ontologists searching for the Holy Grail may find the referenced paper interesting,  esp. wrt events.

Lera Boroditsky (@leraboroditsky) tweeted at 3:03 PM on Mon, Sep 26, 2016:
Is 'eating' universal? In Maniq language, you hãw rice, kap meat, lik mangoes, paŋ yams, and hop soup. Cool paper:
https://t.co/biPLk74Iba
(https://twitter.com/leraboroditsky/status/780498004511956992?s=03)

Get the official Twitter app at https://twitter.com/download?s=13

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joseph simpson

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Oct 1, 2016, 5:23:10 PM10/1/16
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Bruce:

What would you call an N dimensional H-Tree?

What would be the constrains on the N dimensional configuration? 

Take care and have fun,

Joe

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Bruce Schuman

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Oct 1, 2016, 6:59:37 PM10/1/16
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Kinda hard for me to visualize that – maybe I has to start from a point – and branch….  ?? how….

 

So every one of those T intersections is – 3-dimensional ?  or –

 

How does an n-dimensional space branch in all dimensions from a common point…

 

?

 

What the VLSI people seem to be thinking – is that chip design is something like 2.5 dimensional – because the chip has some actual thickness – it’s not just defined in a plane….

 

 

 

Bruce Schuman

www.origin.org/book.cfm

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Joe Simpson

“Reasonable people adapt themselves to the world. 

Unreasonable people attempt to adapt the world to themselves. 

All progress, therefore, depends on unreasonable people.”

George Bernard Shaw

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joseph simpson

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Oct 1, 2016, 9:46:48 PM10/1/16
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Well ... you might think about N parallel 2 dimensional planes that are connected in some fashion...

The H-Tree could be used to organize each plane...

And other concepts (including the H-Tree) could be used to integrate the N parallel planes..

Or something like that... some type of architecture that allows cascading dimensions..

Take care and have fun,

Joe

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Joe Simpson

“Reasonable people adapt themselves to the world. 

Unreasonable people attempt to adapt the world to themselves. 

All progress, therefore, depends on unreasonable people.”

George Bernard Shaw

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Pat Hayes

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Oct 2, 2016, 11:48:45 AM10/2/16
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Come on, people, do a Google search first! 

This was on the first page from the search "fractal H-tree dimensions":

See page 9.

Pat Hayes


<image002.png>

 

“The H-tree: a 2-dimensional hierarchical self-similar branching network that distributes clock signals on an integrated circuit”

 

<image003.jpg>

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joseph simpson

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Oct 2, 2016, 12:10:38 PM10/2/16
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Pat:

Thanks for the pointer to this additional information...

These forms provide the basis for other interesting configurations..

No need to let a Google search limit your imagination...

Take care and have fun,

Joe
Pat Hayes



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