space-filling and fractals have had my fascination since I was a kid… well, late teens when I got a hold of FRACTINT in 1989, downloaded over a 300 bps modem hooked up to PC-Link, predecessor to AOL.
The year after I was in college for a bit and saw the art on mathematics and computer science professors describing chaos theory, others with complexity theory, more fractals, and I *really* wanted a part of all that.
Didn’t happen. ($$$ ran out – thanks Reagan – who counts a house as an ASSET? oh.. they did for a short time) – but my fascination never ended.
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MIT OpenCourseWare was one of the greatest general public good they’ve done. So much great education at MIT and so many good things came out of it. I’m not one to go through a course from START -> FINISH as my brain works in reverse with that stuff. [start with complexity and decompose as I go, stopping when I get ‘stuck’ zooming to the smallest level “building blocks”, and then when I get a handle on the confusing piece, zoom back up again.
Wolfram’s work has been fantastic. The syntax he uses is *very* very similar to Microsoft Excel, which I’m quite familiar with [it’s my sketchpad for everything] – and considering the low-mid entry point for Excel functions comprehension, I can see that Wolfram’s syntax being similar also carries over that same straightforward approach.
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Same. He took Turing/Church through “Game of Life” and made a reality stitching machine. Brilliant stuff.
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He’s very accessible too. Friend him on FB and he’ll add you back.
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Did you get nearly schoolgirl giddy when he accepted? I’ll be honest… I did and I’m not usually much for superstars, but I guess it just had to be a superstar in the right fie
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I love that he had a great idea and stayed with it. Keeps with it. Builds on it. Runs everything through it. He’s enthusiastic about it, even to the point of sounding absurd at times. [there’s times I went, “Oh no Stephen, you didn’t go *there* did you?”] – and I love that.
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I don’t have too many on that level. Wolfram is there. I don’t have Kurzweil there because I’m still seeing him as the guy who made that awesome sampling keyboard and remember seeing him writing about singularity stuff in Usenet groups in the AI groups, although I don’t think it was called that then.
Unfortunately, I put him in the category of “brilliant musician/engineer who thinks he’s got a theory of everything” – that just happened to get himself positioned in a place where he can have an audience.
Hard to shake that prejudice but at the same time, I’m REALLY glad he’s where he is doing what he’s doing.
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But I regularly take down my heroes so I don’t get too deep. I’m working on my own ideas and don’t want *too* much undue influence 
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Nice! Do you want critique if I think I’ve found something [just so you know what potential things you might hear from others?] – or would you like me to just sit back and enjoy and discuss ramifications?
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I’ll let your writing take me on a journey and see where I end up then rather than having a predefined course.
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life always interferes (hence delay) but I saw the overview sheet and it’s impressive. It’s precisely the kind of thing that needs to happen. It reminds me of what game makers have had to do in creating artificial worlds as there can’t be gaps yet everything must ‘fit’ together. It’s a more proper kind of graphing for many purposes but what’s nice about your approach here is maneuvering *all* of mathematics to take advantage of this system. One fantastic benefit I can see of this is it gets rid of the need so much of the bizarre artifacts of going from into transcenentals so often because *so much* relies on the definition of a circle. Plus with our more robust computing tools that *much* prefer dealing with whole numbers such as nodes and edges and routing, you’ve just simplified the whole process of calculating so many things. Example of interest to me would be A-to-D conversion. [which is what this is ultimately about it seems]. The Fourier transform becomes a trivial affair for most purposes if you have waveforms constructed of sin/cos/etc of this structure as unfolded novel circles rather than standard circles. Yeah, I like it a lot. I’ll read the othe rpapers now but the overview seems like precisely the kind of thing the world needs, especially of engineering, but I could also see this of great use in the purely theoretical, allowing much easier to calculate models in theoretical physics, perhaps getting them over the hump of a number of issues. and as the users *know* it’s not intended to correspond to reality but as a precise countable approximation, well, yeah – you’ve just changed everything.
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t’s pragmatic as hell. Suits my way of thinking too as I’m used to dealing with pixels/voxels/bitmapping mndset.
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What’s nice is you have no lack of supporting documents proving the validity of this methodology. Hexagons on a 2D surface are a superior graphing method for many applications already. [I have a free DXF / CAD drawing program I use sometimes that allows for easy work with them]. So much of the work on the pragmatic side of things has been done through the years (as hexagons on 2D surfaces have proven their worth over and over again) that an paradigm shift like this *really* is due.
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Plus, considering that you rarely deal with a lone sphere in a universe of nothing else but rather with packed spheres, why are we going through all this extra work with spheres all of the time?
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Modeling ideal gases just becomes a game of counting now. This in effect turns randomness on its head as well. Statistical analysis then becomes a trivial matter as well. It’d be so nice dealing with wholes again and fractions that make sense.
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Do these shapes pack themselves into other same shapes as you go up scope? I guess they would wouldn’t they? That means you can model the behaviors of subatomics far more easily, once you get correspondence with equivalent wave patterns down pat. It won’t model *this* Universe precisely but neither does what we have now. It’ll allow them to move forward at a much faster rate in an easily computed way.
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You’ve just killed the stupid infinity of triangles which I always hated. Now they can be simply counted because you’re not expecting ‘smoothness’.
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I tend towards preferring “continuum” / infinitesimal thinking because I think it corresponds better with reality. But I’m fine with a methodology that allows for counting so long as it scales up properly. I think this scales up.
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https://en.wikipedia.org/wiki/N-flake#Hexaflake
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My brain’s always going to the land of paper bending into itself, toruses, gravity wells, Kleine bottles and things
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I’ll step back to trig for a bit, sure. As I tend to stay at the level of mental visualization of processes rather than “how to put them on paper”, I forget a lot of terminology.
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Ah! From that book I have my hex-in-a-hex mapping answer.
ok ok I see your point – the graphing while appearing to be hexagonal is actually representative of higher-than 2 dimensions, yes?
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Sort of. Q – Bert lives in a supspace of a hexagonal version of taxicab space.
heaning he can only exist at an intersection.
Ah yes – taxicab space. It’s coming back to me. [did a quick wikipedia – the internet *is* an extension of my long term memory storage]
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” q bert space is discrete and he actually lives on a single slanted plane (Which is a subspace of whatever q bert hexspace is)”
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The fact that I entirely understand that sentence amazes me.
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It’s like a tangent cutting through a circle and creating its own plane distinct from the sphere but intersecting with it.
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a tangent line is a subspace of the plane containing the circle, yes.
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Oh I’m familiar the inifinities of subspaces although I didn’t know which branch handled that. but believe me I appreciate the point-by-point corrections as they come along. [I forgot the def of radians, I mean.. jeez]
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yeah – had to water a plant, help someoen whose back went out get to the bathroom put food in the oven – but thankfully the store-and-forward nature of these things means the msgs will wait
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love it. Started with CB radio for me (my first introduction to half-duplex – what a pleasure that was. One thing at a time, taking their turn.
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k – looking it it now (made 3 copies – yeah I’d swear I’ve seen something similar to this before for mapping 3D volumes digitally but it’s not coming to mind just yet. Ok, I have pencil.
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but wait – i didn’t really end up at origin did i?
because I went up 3 in one dimention up 3 in another, up three in another
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THe vector is < 3 3 3 > the corresponding point is ( 0, 0, 0)
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If I did this for needful things, I’d probably switch pen colors Continue.
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Ah! because you’ve just made a a shape that projects to a triangle in 2D but isn’t. It’s that famous optical illusion shape.
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Well for now, that’s my mental visualization until corrected.
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<4 5 6> corresponding to -1 0 1 on the paper.
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no i got it – it was task shifting and i missed the shift – differnet mode of thinking – [now I have to look down at the paper from above, see the whole map, find the point, then travel] – ust a moment
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5 spaces but they go down three in one dimension, up 2 in another dimension.and remain in origin dimension for 3.
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[I had to spell it out in words so I could keep the mental shape in mind]
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and now add the vector < 1 0 0> then add < 0 1 0 > then add < 0 0 1 >
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kk. I am going to head out for some dinner too, but I’ll be online.
< 1 0 0 > + < 0 1 0> + < 0 0 1 > = < 1 1 1 >
the first three are the A COMPONENT VECTOR, the B COMPONENT VECTOR and the C COMPONENT VECTOR respectively of the RESULTANT VECTOR < 1 1 1 >.
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ok – everything has to stay in its proper slots. Yeah – forgot that [were it Excel, I’d track the in different columns]
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Not sure whether it’s called a syntax or semantics error or something else, but its an issue I run into when dealing with notation systems sometimes.
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So, 1 1 1, 2 2 2, 3 3 3, 4 4 4 – I can see how powerful this becomes.
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if you go the same number for each dimension on the paper you end up in the same spot on the 2D graph.
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“his space can be thought of as the projection of cubic euclidean space onto a horizontal plane. but please don;t pay much attention to that.”
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< a b c > + < n n n> = < a b c >
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It has multiple correspondance – one-to-many relationship.?
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It’s only limited to the size of the paper. If the paper is infinite in both directions, it can describe any point along the projected 3D space represented by that single point. So, each point represents an infinite space in 3D is that right?
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it may describean unfinite space in some sense. but the point i am making is that there are an infinite number of ways to represent every point.
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so understand that while there an infinite number of ways to describe each point. all of those ways describe the same point.
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That’s the nature of projection though isn’t it? Shadows projection on a wall are can represent objects in 3D space at an infinite distance away, in a theoretical universe where you didn’t have to worry about certain other things.
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Maybe I’m thinking of translation. Like, on my monitor, point 2,2 can represent a virtual space. The limitations of what that 2.2 can represent of that 3D space is functionally infinite.
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Yeah – I remember messing around with doing these calculations in BASIC to make a little tunnel I could walk down and turn in. Made me feel like a god.
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The pixels on your paper monitor are hexagonal.
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But they’re not referenced in 2D coordinate system but 3D coordinate system
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You’ve written a coordinate system that would be at home inside of a graphics card that can do matrix manipulations natively.
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I’ve been messing with computers since I was 11. Now 45. I was going to go into theoretical physics in college with high recommendations from teachers but the professor I wanted (who had an intro “Quantum Mechanics for the Myriad” during an orientation course) was on sabattical. So I did cihld psychology/programming instead but still didn’t finish).I always had the intuition for it just not the $$ for the education — and thank you
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Been here since ’89 and haven’t left. I was at the least structured college (Hampshire College – no grades or tests) – and it still wasn’t unstructured enough for me)
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The flaw is the structure of academia. The principles it operates from discourage the kind of thinking I do, even in the most creative colleges. hampshire was the closest but even there, there were prerequisites. Internet is for me.
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That’s how I focus. I go lateral then come back. I incorporate all of me into each item, rather than creating an artificial isolated universe where the only thing that matters is in front of you.Now I’l return to you –
You chose them because they’re relative to a central point that isn’t really central.
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Gotta start somewhere. What’s nice about your system is that the origin point is of convenience not necessity. Any point is as good as another so it has a nice combination of absolute and relative to it. [because once you leave 0,0,0, even if it seems like you’ve returned to 0,0,0, you *have* returned to 0,0,0 and yet you haven’t because you’re now off in 9999,9999,9999 land.
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My mind-reading abilities are shy so you’ll have to help me as to the more important relevance you see
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The way I see it, from the comfort of your pencil and graph, you can cover a lot of space (as long as you keep a tally on the side as to how high you’ve gone)
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As long as you’ve been keeping track of where youve translated up to, then yeah.
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I’d probably add some columns off to the side in order to keep track of the point shown on paper and the corresponding point you’re working with, since each point can represent any number, even if they’re side-by-side on the paper.
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t’s a straight line. You’ve turned circles onto their side and turned them into lines.
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t’s a hexagon on the graph but what it’s representing isnt’ a hexagon, at least not in 3D space.
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that’s more what I expected to see.
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of each *row*? ok, this is what distraction everywhere gets me. ok. I’ve got the list and summing up the absolutes of each row (abs(x+y+z) down the row = 30. Unless you wanted the absolute of each value first, then summed row then summed column?
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ah, you’d have to stay relative to 0 0 0 in order to get back again.
So, I got 30, which makes sense. hexagonal math is one of the oldest out there, source of our seconds, babylonian and stuff.
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“The sum of the abs of the chords is the distance from the origin”
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and, being a perfect coordinate system, you don’t need breadcrumbs to get back to 0 0 0 as you would navigating a cave or doing a random walk.
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