What is this math phenomena?

edgeofinnerspace's Avatar

edgeofinnerspace

06 Feb, 2024 04:33 AM

Hello, apologies for the absence my dad had a stroke... as for the rest of it, I'll spare you the details John.
However, here is something that has me intrigued - I know there must be some name for this right?

I have set the "min" and "max" on some ports just so you know what mine were set to when I first notice and so it's easier to see what you are adjusting at first to see what I'm describing.

-Create a triangle of hexagons that touch one of each adjacent hexagon's vertices. (...or approximately, in our visualization - I don't think it matters)
- Slowly increase the radius (bottommost setting) until one vertex from each hexagon intersects at the center paying attention to shape formed in the middle of the of group as a whole. Keep going and the the vertex of a neighbor and a hexagon opposite will cross another and then until the last vertex of the last vertex of the last hexagon - whichever you have looked at throughout this process.

Now at this point, the shape in the middle has gone from a star, to an equilateral triangle, and now it is approaching and ends as a hexagon with the distance from a "side" of the hexagon created by the overlapping of the original smaller hexagons to the next line and the next, etc, traveling outward as if drawing the apothem and continuing past the side of the middle hexagon we created.

But how is that possible that this outer bound (of the hexagon created in the middle) ends at that "constant" ...value... I don't even know what it would be called. Everything appears dynamic until that point where some type of "stability" for lack of a better word has been reached.

What's up here? Where is that called?

This is why, other than my own self you are the greatest resource I have - because usually when something like this distracts me, I am avoiding something important I need to do. And being my own greatest resource, if I am stuck out of curiosity created out of my own procrastination, having a resource like you to settle and satisfy these types of curiosities is imperative. I would go on for hours searching for the answer but I know in this case who would know or atleast know where to look! And I doubly know google or others won't know jack when John does.

  1. Support Staff 1 Posted by john on 06 Feb, 2024 12:06 PM

    john's Avatar

    Dear Edge,

    Good to have you back on the forum! Things were getting a little too quiet lately.

    First, my condolences about your dad. Sounds like you and your family have had a rough ride. I hope you are finding some peace and healing.

    As for your "hexagon puzzle", there's not really any "math phenomenon" going on here, just the effects of relative scale on our visual systems.

    As you increase the radius of the hexagons, at first the interference effects of the edges are large compared to the size of the nine hexagons. They seem to form rapidly changing patterns in the center. This is a bit of an illusion. There are always just 9 overlapping hexagons, but when they are relatively small our visual systems pull out striking patterns - like a six-pointed star or a triangle - that are actually formed out of selected bits and pieces that we emphasize in our minds.

    But as you continue to increase the radius past 150, the true nature of this design becomes apparent: 9 overlapping hexagons, slightly offset from each other. When the radius becomes much larger than the amount of the offset, the space near the center empties out and the only interference happens near the edges.

    This offset remains constant throughout. It is defined by a value inside your move sub-subnetworks called gridStep. In the version you sent me, gridStep is 50. That is, the centers of the nine hexagons are 50 pixels apart (measured on a triangular grid). And because the centers are always 50 pixels apart, the vertices of neighboring hexagons are always 50 pixels apart.

    All of this becomes clearer when you show the centers and color some of the overlapping hexagons. I modified your network to do just that (screenshot and modified zip file attached).

    The 9 center points are shown as a triangle of dots. Most are black, but I turn the first one red and another one blue. I then show the corresponding hexagons for those two centers in translucent red and blue. Finally, I add measurement lines, one between the red and blue centers, another between northeast vertices of the red and blue hexagons.

    The best way to really grok what's happening here is to slowly increase the radius and watch what happens to the colored hexagons. I made a quick movie showing this effect (also attached).

    I hope that clears things up. Our visual systems are creative and are good at seeing patterns that may not really be there. The "stability" that becomes apparent when the hexagons get big enough was really there all along.

    Thanks for your other notes and please keep posting!

    John

  2. 2 Posted by edgeofinnerspac... on 21 Feb, 2024 04:49 PM

    edgeofinnerspace's Avatar

    Ok that makes more sense now. I'm posting an even more interesting one now then. It time for cymatics in nodebox.

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