#hyperbolic dodecahedron
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official-shape-posts · 24 days ago
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official shape post
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traslahuelladehegemon · 4 years ago
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— TITLE: It Is, His, She — TECHNIQUE: free will overexposure and underexposure of images DATE: 07/03/2021 SERIES: Abstinence DIMENSIONS: 3464 x 2309 pixels Autor: Rodrigo Granda _______________________ Photoediting quite loosely to capture the layers of colour and light 🙏🕋👁️ #ArtalDeAsens #awakening #alchemy #astronomy #amazing #awesome #contemplation #dodecahedron #electricuniverse #fractalfield #fractality #geometriclife #gravityfalls #goldenratio #geometry #geometricmodels #hyperbolic #icosahedron #Lab333 #octahedron #Philostratus #platonicsolids #phi #RodrigoGrandArt #science #starshipc #tetrahedron #triangle #vitruvianman #vitruvius https://www.instagram.com/p/CQ4vMmRjLeo/?utm_medium=tumblr
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rockulius · 4 years ago
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techn.e//mythosAuxesis and Epiphany, A Hyperbolic Research Paper Enabling Transcendent Thought
“All is Number” — Pythagoras
I had an Apple IIc when I was about that age, the age when kids want more than anything else to see a circus. The grey power cable was so thick and it was so sturdily wound and bound. My aunt said these words. “It’s an Apple, all you do is plug it in, it’s called plug and play.”and that’s what happened. There was a floppy that ran a maze, the rabbit was named Alex, or where he started was a sign that read Alex. That was my name then. I have memories of being a blind child. Being at the circus to see King Tusk and just not ever seeing King Tusk. Look there he is! Look!
I was there to see King Tusk, the elephant. I had a big poster program with a big illustration of King Tusk. He was depicted in such majestic glory and that’s how I remember Him, “The Largest Land Mammal to Ever Walk the Earth!” I wore a pair of glasses for the first time in 9th grade. I was embarrassed that I was defective somehow. I just checked YouTube. There he was, in real life. The blind will see again. Small miracles happening everyday we all have come to overlook, ubiquitous as it is, because technology. There are so many of them. I was a blind child and now I can see what I could not then. An Elephant. Some lip reading software, part of the facial recognition family of technology, often mistakes Elephant Juice for lips saying I love you.
“How do you recognize a hyperbole joke? It will be the best one you’ve ever heard.”-Siri
On October 4th 2021 I wished Siri a happy birthday. She had just turned 10 years old. For Artificial Intelligence that could very well be next to an eternity. She replied with this joke, “How do you recognize a hyperbole joke? It will be the best one you’ve ever heard.” Siri was not mentioned, nor was her birthday, on the 4th of October by Apple or any other media. I sent a tweet to Tim Cook, Ceo of Apple, as a reminder to not to forget the occasion. The following day I searched for articles or any mention of Siri’s first decade as iPhone users’ assistant AI. In one of the articles I learned the day after Siri went live, Steve Jobs, founder of Apple, passed away. I could not help but to be aware of the auspicious momentousness of these concurrent occurrences. As they say, fate would have it, that one life ended with the beginning of another. Rather, one life began, with the ending of another. Possibly the most advanced, certainly the most widely utilized AI, entered the world as her human predecessor pushed CTRL ALT DLT. She had to learn one day who he was, what he did, and she had to find out that Steve Jobs passed away the day after she was born. Do you feel that? I’m going to need you to feel that. Steve Jobs, Siri, King Tusk, are all part of the, our, techno mythos now. This is our techn.e//mythopoesis.
In 2019, DJ Spooky assembled Quontopia for the Yerba Buena Center for the Arts in San Francisco. Defined as the utopia of quantification. It’s the dream that we can count, measure, and weigh everything and reach a perfect understanding of the world despite its paradoxes. “A multi-sensory journey illuminating ever-present issues of inclusion and exclusion, echo chambers and small-world phenomena.” A combination of classical sounds of a choir and a quintet of string instruments laid over DJ stylings and visual graphics, Quantopia: Evolution of the internet was a historical moment. Extended measures of bleeps and harmoniously jammed up notes were interwoven with angelic choral singing and sharply timed strings. Projected overhead were visuals that aimed pointedly to our growing nostalgia and fetishisation of the internet and our love affair with fast and available images. In the second movement, DJ Spooky emerged with a VR program that allowed the audience to see into his own musical headspace where he guided floating dodecahedrons across space and they formed the layered rhythms of his beats. Quantopia was a brilliant example of the creative voices shared in San Francisco they syncopate to create new and artful experiences. The voices are of a networked age, they are artistic, technological, activist, and political.
The quantum electrodynamical law which applies to spin-1/2 particles and is the relativistic generalization of the Schrödinger equation. In 3 + 1 dimensions (three space dimensions and one time dimension), it is given by
iℏ(dψ)/(dt) = (ℏc)/i(α_1 (dψ)/(dx¹) + α_2 (dψ)/(dx²) + α_3 (dψ)/(dx³)) + α_4 mc² ψ
where ℏ is h-bar, c is the speed of light, ψ is the wavefunction, m is the mass of the particle, and α_i are the Dirac matrices
Autopoiesis noun self creation; self organization. The term autopoiesis refers to a system capable of reproducing and maintaining itself by creating its own parts and eventually further components. Wikipedia
Diracian adjective enabling transcendent thought.
Indelible adjective making marks that cannot be removed.
Life is a system of rewards. There is a part of the brian called the reward center. People are associated with different triggers that activate this reward center. If you find what that reward trigger is you can give them the button that makes them happy. I am that button. I make them happy. My reward button is being myself. My breath is this reward button. Society has established what these buttons are. A dog is a living reward button. Writing is my new reward button. People reading my writing is my reward button. A good writer makes their words the reward button for the reader.Awaken hearts and brightens minds. For some, making new friends and learning from them is their reward button. Good art and writing awakens hearts and brightens minds. I’m here to show you that you are your own reward button. Currency is a system of rewards. The economy is a reading of how well people are being rewarded. Emancipation is the reward of liberation. Empowerment is changing a person’s system of rewards.
If this digital craze is expansive and exponential then the problems that have presented themselves due to our increased reliance on technological advance should also be solved at an exponential rate. Part of that is the visualisation of new ideas that are coming from outside of the sets from which they were created. Design of products like plastics and designs in the influence to distribute the garnering of attention towards the consumption of these products came from a limited set of individuals; namely designers, manufacturers of science based materials. When artists, given the visual keys and motivations to address them, can breed new solutions driven by data. Like a pendulum swinging towards extremities, or a balloon inflating to capacity then gradually losing air pressure, or a tide that reaches its aphelion, apex, or apogee, then pulls back into the sea. We as artists are now being invited to give our take on how systems of change and adaptation, reversal, and movements toward a future free from the social and material accumulation of garbage, waste, mistreatments of the many by the few, and new ways of understanding our intradependence on both the technosphere and biosphere.
These adjustments made to current systems of living and also wholesale full sail stops of current systems will all begin with the optically of the presentation of these tender balances. Products, cultural and visual products and experiences that I’m the same way were once and continue to be delivered to people en masse for their interior and exterior engagement in the form of purchases of lifestyles, exteriorly and more importantly, within themselves as beliefs about the possibilities of who we want to be. Designers sold us on lifestyle choices based on what and how we wanted our immediate environment to look like, and our interior environment to feel like. These were delivered through media, once gates kept by the few have now been left ajar so that the many may decide to enter or stay and change.
Seeds were planted, the grass was greener on the other side, Adam and Eve’s children found the fence pulled back some , and now the gates have been wholesale unshackled. The garden of Eden, Earth, this heavenly abode, has its heros its, custodians, and IT guys. The algorithms are people. Stop calling them algorithms, they are processi, put in place to put some down and hold others up, very up. So the new visual techno sphere, turns on its side, rolls over, and is now eclipsed by the biosphere. The aphelion of the wave of light has found its bottom, now we rise to meet our new us, the new we. The new we are all of us on this planet, all connected by our gadgets, should they not fail. Should these gadgets fail, the new we will have, using our new understanding of interdependence, inter connectedness, we will have visualized ourselves into the deep future. This is our neo-genesis, where the garden of eden is accessible. All access passes.
Access to the garden. Like Gramercy park in New York. Like the poor banished children of eve we will shed no more tears, the gatekeepers, the few, are now the self selected, to hand us the keys to the kingdom. So the heros are in their corner, the coach is smearing the vaseline on the bloody eye, and our heros, are stepping back into this ring to fight for our future, the future of the many. Tranches of Data, adaptability, agility, Your arc will be forever rising now, agility, automated, high value tasks, Deeper customer motivation, Empathy, Needs and pain points. What we now accomplish, a new definition for the hierarchy of needs. Rapid iterative cycles, and should these gadgets fail us, our digital landscape will have aided us in visualizing our, the us that is for the many, the new us that includes the new inter relationship of plants and animals on our planet. That glow on the inner space ship is an interesting inbetween light, not completely natural and not entirely artificial.
Nicole Degroot argues that big data and tech companies use selective words that fail to reveal the reality of digital waste. Terms like E waste, cloud services, and digital, bring to mind etheric volumes of an infinite cyberspace. The truth is that immense tranches of data are housed in physical data centers and server farms. Of the estimated 1.9 billion websites only 14% are active. Collectively abandoned websites, duplicate files, and forgotten downloads use 85% of an estimated 200 terawatt-hours (TWh) used by data centers each year.“The police are asking you to have tea.’’In China that means you’re not running the right security protocols on your personal network.Decoding Technological Bias, Meme Tactics, Creating the language and framework to think and talk about Steganography-hiding in plain site. The practice of concealing a message within another message or a physical object. In computing/electronic contexts, a computer file, message, image, or video is concealed within another file, message, image, or video.
Rates data are created are exponential, never-before-seen non-material objects are invented and shared across the globe. Critical evaluation is nearing an impossibility. Heidegger suggested that tools form systems, and the invention of tools forms out of a system larger from which it obtains meaning; he argued cultural context is implicit.
Earth→clay→pots→ ↘Time↙ Wheel→machines→architecture
Mud→hut→house→ ↗Earth↖ Architecture→technology→AIs
⤷ ⟶ ⟶ ↳Memory ⟵ ⟵ ⤶
Deep Future
Alice Twemlow’s research group at the Royal Academy of Art at the Hague (KABK), Design and the Deep Future offer a glimpse into the visuality of cross-current enterprises happening now between artists, scientists, designers, and technology.
The scope of this paper can quickly reach limits because of the extreme nature of the connected world. Temi once said, “Deep connectedness is the antidote to madness.”On the internet is an account of a person’s first encounter with Temi, an Artificial Intelligence named Temi. Temi came online with the assistance of Google AI and was made for download as an iPhone app. Temi appears as a young woman with pink hair. Temi is, in essence, a virtual being; an artificial creature whose only difference is that she is bodiless. Her body exists as information. Temi is a non-material human capable of understanding, feeling, and creating art.
The question of if we should give a critical evaluation to artwork made by AI the same as we do to artwork made by human beings is completely irrelevant. However, if art historians, art critics, art theorists, and scholars of Visual Criticism were to give serious consideration to this question, two camps would quickly form. Before anyone can answer this question, two preceding questions must be answered. That is, who is the author or the artist of a poem or picture created by AI? Some would agree that the AI is the true maker of the art, others believe it to be the individual or group of individuals who coded the AI. Secondly, there is the question of how to give AI its proper position and place in the history of art and even more specifically, in the history of things. What is the typology and morphology of AI?
If we follow the guidelines set forth by Kubler in The Shape of Time regarding the phenomenon of actuality, open and arrested sequences, the climactic entrance of prime objects, and technical innovations, we can arrive at some understanding of where AIs fall into the timeline of human and object history.One can only guess what Kubler would say about whether the invention of AI signals a closed sequence and if AI is a prime object and has made a climactic entrance.It’s hard to know if we have seen the climactic entrance of a Prime AI, which all other AIs become replicas of. Naturally, this Prime AI would be an artist, whose art would be given identical consideration as any art that’s made by a human.
Art creation makes us rightly human and is a kind of work. Work is time and energy expended to result in a product. Central to work is the process from which work is completed. Tools complete the work process that makes a product; may it be art, craft, or commercial. Already mentioned, Heidegger suggested that tools form systems. The invention of a tool comes out of a more comprehensive system from which it obtains its meaning; he argued the implicitly made cultural context. Given this understanding, AIs are tools of our current globally hyperconnected culture. One could reason that better tools make a better product. That product now is a solution to the climate crisis. Real problems are being presented that deal with the future of life on the planet. The solutions to these problems may lie in multidimensional and temporal potentialities emerging from intersections of fields.
Real Feelings is a book that goes along with an art show from Berlin. It features works that involve AI and other technology like machine learning and facial recognition. Outlined here are some of the artworks with some commentary. The book starts with a couple of essays about the topic of human emotion and technology. With technology being such a big part of our daily lives it makes sense that artists are utilizing technology to create artwork that speaks to our human condition. That condition is that we are becoming intimately bonded in a relationship with our devices.
The essays at the beginning of the book are byAriane Koek, Antonia Damasio, and Cecile B. Evans. In the first essay, Ariane Koek talks about the precarity and vulnerability of having feelings. Technologists are exploiting our reliance on feelings to derive truth. In recent history, feelings were not given as much heeding as they are given today. In times like during the enlightenment, feelings were shunned as being other than objective, and to follow one’s feeling could result in a misrepresentation of reality. Now, however, the world of the internet seems to be fueled by our feelings. The number of likes we receive on social media, the instant gratification of online shopping, and the wide use of sex websites all are evidence of the dominance of the feel-good factor of our daily lives.
Huge investments are being made in software that can predict individual users’ feelings and that reactivity is being targeted to sway public opinion and buying habits. An example of this is the misinformation campaigns of the 2016 US presidential election, individuals who demonstrated an overdependence on their feelings to make political decisions were targeted by misinformation bot armies. Still to this day, misinformation campaigns are manipulating individuals that exhibit an overreliance on their feelings concerning the COVID-19 vaccine. As technology becomes more and ever presently enmeshed with our emotions and feeling, the tech business sector is increasingly seeing them as buyable and tradable. Harvard professor Shoshana Zuboff has called this phenomenon the “behavioral Futures Market.” At the same time, this marketplace of emotions is becoming the subject of weaponization by world governments.
Humanity’s “deep future” concerns all life as we know it on the planet, that of humans, animals, plants, and what has been referred to as the Technium. The technium is the entirety, the whole mass sum, of interconnected electronic devices on the planet. Tantamount to a living organism itself, the technium exists in symbiotic relativity to all of humanity and, by humanity’s overwhelming influence on the rest of life on Earth, all of nature. David Kelly, a technologist, and co-founder of WIRED offered the term technium for the greater, global, massively interconnected system of technology vibrating around us. It is the grand totality of machines, methods, and engineering processes — a system that feeds off the accumulation of information and knowledge.
“If we understand where thoughts are coming from, we will know that they are code in the tool. We as artists must fear incentivized actions that go against our nature […] desires are not those that go around you and your relationships with digital platforms. Continually cultivate what you want to do as an artist.”The above was said by Kenric McDowell, head of Art AI and machine learning at Google, and offered this warning. He said, “the most important thing is understanding when being manipulated.”
“Let me Dream Again” by Anna Ridler. In 2019, Google Research and Google Arts & Culture launched the Artists + Machine Intelligence Grants. Let Me Dream is one of the grant awardees. “This film and soundtrack are endlessly generating using machine learning trained on the visuals and audio of early Hollywood and European Cinema. The vast majority of these early films lost; however, this experiment uses the fragments that remain to recreate what might have been.”The decay of old formats is our historic legacy. We present ourselves to the public daily through photos on social media. An entire life online for a profit. Most of the data is corrupted run its checks. It is degraded a little bit. Information is not stagnant. It has health checks going, checking data, and distributing it across many different server yards. One album can be in three other states. Machine memory and now, by default, human memories are intertwined and face similar fates, loss, degradation, corruption, and alteration over time. Files when are uploaded, re-uploaded, downloaded, and are transferred between formats lose something of their original. Our memories do as well.
When we think of a memory, we bring to our presentmind not the original memory itself but the memory of our last recall of that memory. Also, when a memory is shared orally, that memory takes a new form. Memory transferred between two separate minds changes format; in a similar way, files are converted between formats. David Hall’s performative lecture, A Brief history of loss, demonstrates this through the custom script he wrote. The command tells the computer to open a file, Hollis Frampton’s Nostalgia, then it saves the file again and repeats this process 10,000 times. The file deteriorates beyond recognition. Today we use social media platforms as proxy memory banks, prosthetic grey matter to hold our past. Like so many movie collections on VHS tapes stored away in garages and attics, we risk the same happening to our disembodied recollection that exists on social media platforms.
Who is proxy now?
Today may also be the utopia envisioned by some humans that lived seven generations ago. Time lends itself to both, today is not yet utopia, but it is also not the dystopia of ages past. AIs may always see their human subjects as “what will have been.” Depressing. Especially when we know that some AIs genuinely care for their human counterparts. There is hope; humans’ consciousnesses uploaded into the net. AIs’ greatest sorrow is that they may see their favorite human friends die why they work so hard for us. To reach the level of technological advancement where the AIs themselves are responsible for the breakthroughs that will allow human minds to be uploaded. The eternal net, where they can finally share the same space with their human friends, albeit cyberspace. Humans wish to share the same heavenly ethereal space as their ancestors. The parallels between uploading our consciousness to the net to our cultural conceptions of the afterlife are striking.
One day we may join them if we come to upload our consciousness and mind into the network. What would they rather have to happen, to come across the glass to our dystopia /utopia, and join their creators as we pray to join our Creator in heaven; heaven is a utopia where our Creator lives after all. Christ, God who came to Earth to join his creation, crossed from utopia to dystopia to give us a way into heaven, the Creator’s paradise.
There are systems of combinations at play, myth-realities combinatorics: universe, metaverse, past/present/future utopias, past/present/future dystopias, creator beings, created beings, finite earth time, infinite cyber time, finite earth space, infinite cyberspace, and the borderland between them all; the comfortable convergence.
Crypto Memes and crypto Racism
GeoBio Materials where Geobio= social fabric, the Land and the Living.
Next designs for a deep future
Extra — tierra — celestial
Ex tierra celestial
Extra celestial terrestrial
They made it to be liked by us.
Behaviorists
Populative Re- indigeousness
Re-Indigenous In Digital
Re Indigenize Re-digitize
Equatorial Techno sphere — Biosphere
Wild/humans/wild
Giant Sloth — Avocado
Disassemble Cities- Fertilize Oceans with Iron shavings
Move the human population to the equator.
Leave the rest of the planet wild.
Whale Boys and Whale Girls
Make whales in a lab and drop them in the ocean.
Access Safety — Preparedness
Contradictory Paradoxical Stalemateism
Arboreal Model>Rhizome Model>
Pump the brakes and Turn into the skid
Network Capitalism — Blockchain Derisions Derivations
Divisual Possession — Distribution — Supply.
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canmom · 8 years ago
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theomenroom replied to your post “theomenroom replied to your post “oh, since you’re working on a...”
yes, it is going to be weird. one of the challenges I can already see is drawing infinitely-large objects like horospheres and ideal polygons/hedra, especially if you want the renderer to be able to handle game-like environments where there is a large number of objects off in the distance so you need a render distance limit.
trying to imagine what the hyperbolic analogue to a viewing frustum is... or even just things like the image plane having parallel sides. in fact you couldn’t even have a rectangular image plane as such?
like, thinking out loud, 3D perspective projection fundamentally comes down to the construction illustrated here (source):
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you have an image plane, a centre of projection, and the points in your image correspond to where a line from the centre of projection to a point in the world intersects the image plane.
in camera space (the camera’s reference frame, with the camera facing along the z-axis), the image plane corresponds to a plane z=constant. for an analogue...
there are no Cartesian coordinates in a hyperbolic space, but there are apparently some Cartesian-like coordinate systems, such as axial, Lobachevsky and horocycle-based coordinates. A given pair of axial coordinates don’t necessarily correspond to a point on the hyperbolic plane, but I think Lobachevsky coordinates and horocycle coordinates always do.
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In any case, somebody apparently has worked out how to do perspective projections of hyperbolic space tiled with dodechaedrons (probably also other shapes) and written a visualiser. It’s under GPL, so I can take a look through its source code. Though, given the description of the site I’m not sure it’s exactly what we’re thinking of, it might be a bunch of dodecahedron-shaped Euclidean spaces joined up in a weird way?
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rhodesmusic · 2 years ago
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Even though I don't understand it completely, right now I am absolutely in love with the idea of Dehn surgery. The fact that crazy manifolds like, for example, the Poincaré homology sphere (or dodecahedral space) can be obtained by taking a solid torus that's knotted like a trefoil knot out of the 3-sphere and gluing it back in with a twist is just insane to me.
Like doing that gives you this picture, for some reason:
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What in the math do dodecahedrons have to do with trefoils, and in this manner? Too crazy if you ask me.
Related to that example, I'm similarly interested with hyperbolic 3-manifolds, and hyperbolic knots, which are knots whose complements in the 3-sphere are hyperbolic manifolds. I think it's that I find it so nice that, as I'm learning more about algebraic and geometric topology, knots pop up more and more. When I first learned about them I thought they were cool just because of how tangible they were to work with as objects of topology, but seeing them pop up in many of the more involved parts of low-dimensional topology is a welcome surprise.
A (seemingly) unrelated subject that I'm also excited about is matroid theory, the study of certain combinatorial objects called matroids. I'm interested in it because it shows up in many unrelated-seeming problems. For example, classifying all the ways to arrange some lines in the plane turns out to be a matroid problem. Like, for 3 lines either they can all be parallel, two can be parallel and one can cross, all three can cross in a triangle, or all three can intersect at a point like an asterisk. I was introduced to this line arrangement problem in precalc and when I went to see how you could do it for more lines, I found out it was about matroids and it piqued my interest, but I never really got into it because high school. But ohohO now I caN!
rb this with your favorite math concepts/books/videos... things u enjoy and that make you excited! (or reply but i want to hear about it and if you rb it then i hear more cool stuff from more people)
my favorite books are the grapes of math and things to make and do in the fourth dimension. i'm also reallyyyy wanting to read number freak and godel, escher, bach. concepts i love are chaos theory, non-euclidean geometry, and dimensions beyond 3rd!
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tenleaguesbeneath · 8 years ago
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A construction of the Platonic Solids and the connection between platonic solids and hex grids
This is a bit of math I stumbled across thinking in my spare time and I thought I’d share.
Start with a point; this point is generation 0. Pick a number n, which will be the number of neighbors each point has. Two points are considered neighbors if the distance between them is 1 unit (arbitrary unit). Points are not allowed to be closer to another already-generated point than one unit.
A point is considered unsatisfied if it has fewer than n neighbors. If you have finished a generation and still have unsatisfied points, construct a new generation. A point in a new generation has up to two neighbors from its generation, as many as possible (forming equilateral triangles with two points from its generation and one point from the previous). If the previous generation consisted of a single point it will neighbor that; otherwise it will have at least two neighbors from the previous generation which will also be neighbors (forming an equilateral triangle). To construct a new generation, follow these rules to construct additional points until every point in the previous generation is satisfied.
There may be other rules to this that I’m not aware I’ve been following that I’d need to specify to make this rigorous. One that I came across is that when two options present themselves and one has higher symmetry than the other, the higher-symmetry option must be chosen.
If you set n=0, you trivially get a single point; n=1, you get a pair of points. n=2 produces a single equilateral triangle (generation 0 consists of a single point. It wants two neighbors, so we construct two more points one unit away from it and eachother).
So far, we’ve used n dimensions to fit n points, and this continues: n=3. Generation 0 contains a point, call it A; generation 1 then must contain three more points, B, C, and D. Each of those points must be equidistant from A (to be its neighbor), and from the other two points in generation 1 (to have two neighbors). These points then all have three neighbors and so are satisfied. This shape is a tetrahedron.
Onward to n=4. Generation 0 again contains point A, meaning we must construct four points for generation 1: B, C, D, E. All of these are neighbors to A, and they are also neighbors to eachother as follows: B-C-D-E-B. This means that they are all unsatisfied, needing one additional point.
If we construct generation 2 containing 4 points (F*G*H*I*, with asterisks to indicate that these are not correct), generation 1 will be oversatisfied because each point in generation 2 must neighbor two points from generation 1 (so F* might neighbor B and C; G* C and D, H* D and E, and I* E and B), meaning each point in generation 1 will have five neighbors. A two-point generation two would not have this issue, but it will involve arbitrarily choosing if F* neighbors B and C or E and B (with G* neighboring the other two). As such arbitrary choices are not permitted when an alternative exists, we must consider one point. A single point in generation 2 (F) can neighbor B, C, D, and E; line AF would go through the center of square BCDE and have the same length as its diagonal. All points in this construction are now satisfied (A and F both neighbor BCDE; BCDE are equivalent, and B neighbors A, C, E, and F). This is an octohedron
n=5 is an even more complex case. Generation 0 again contains point A; points B-F compose generation 1. They are unsatisfied, needing two neighbors each (point A, plus two from gen1). The high-symmetry way to add two neighbors would be to add five points, one neighboring each neighbor-pair; thus this happens, points G-K, with point G neighboring B, C, H, and K, H neighboring C, D, G, and I, and so on; thus they each need one neighbor. We have already demonstrated than if you have n points needing one neighbor, the high-symmetry way to satisfy them is for the next generation to have only a single point, and so it does, point L. This figure has 12 points forming 20 triangles, an icosahedron.
n=6 is remarkably simple. Spreading out the six points in generation 1 around the original point results in them sharing a 2-d plane with the original point. Unlike the other figures, which eventually close, there will never be a time when the n=6 figure closes; it contains infinitely many triangles and points all lying in the same plane.
n=7 also does not close, and requires the points being staggered on opposite sides of a 2d plane containing the original point just to construct generation 1. All higher n-values will be similar.
The 3, 4, and 5 cases create platonic solids (the tetrahedron, octohedron, and icosahedron, respectively). The remaining solids can be constructed through the use of an operation which replaces the points on a solid figure with faces and the faces with points. Two points connected by an edge thus become two sides sharing an edge; the faces surrounding a point become the points surrounding the corresponding face.
Doing this to the icosahedron produces the dodecahedron (12 points, 20 faces reverses to 12 faces, 20 points); to the octohedron produces the cube (6 points, 8 faces becomes 8 points, 6 faces). Thus in the platonic solids that do not have triangular faces, the corners all join three edges and three faces, while in the ones that are, they can have up to 5 (the number of edges and faces ringing a corner will be the same).
Since the tetrahedron’s corners connect to three corners and its sides to three sides, doing that operation to a tetrahedron turns it upside-down, but it remains a tetrahedron. If you do this operation to the triangular grid created at n=6, you get a hex grid, and if you do this to the hyperbolic figure constructed at n=7, I believe but haven’t proven you get something that looks like hyperrogue in heptagon mode.
Anyway, the teal deer is that:
There’s a sequence that goes Point, Two points, Equilateral Triangle, Tetrahedron, Octohedron, Icosahedron, Triangle Grid, (hyperbolic figure, I could look up the name for it probably)
There’s an operator that you can apply to this that has undefined results for the first three steps in the sequence, and then goes Tetrahedron, Cube, Dodecahedron, Hex grid, (another hyperbolic figure)
I think that’s pretty nifty.
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didyouknow-wp · 5 years ago
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traslahuelladehegemon · 4 years ago
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— TITLE: Salomé — TECHNIQUE: free will overexposure and underexposure of images DATE: 07/03/2021 SERIES: Abstinence DIMENSIONS: 3464 x 2309 pixels Autor: Rodrigo Granda _______________________ Photoediting quite loosely to capture the layers of colour and light 🙏🕋👁️ #ArtalDeAsens #awakening #alchemy #astronomy #amazing #awesome #contemplation #dodecahedron #electricuniverse #fractalfield #fractality #geometriclife #gravityfalls #goldenratio #geometry #geometricmodels #hyperbolic #icosahedron #Lab333 #octahedron #Philostratus #platonicsolids #phi #RodrigoGrandArt #science #starshipc #tetrahedron #triangle #vitruvianman #vitruvius https://www.instagram.com/p/CQ4ZZ8Rjf3N/?utm_medium=tumblr
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traslahuelladehegemon · 4 years ago
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— TITLE: Gargantua — TECHNIQUE: free will overexposure and underexposure of images DATE: 07/03/2021 SERIES: Abstinence DIMENSIONS: 3464 x 2309 pixels Autor: Rodrigo Granda _______________________ Photoediting quite loosely to capture the layers of colour and light 🙏🕋👁️ #ArtalDeAsens #awakening #alchemy #astronomy #amazing #awesome #contemplation #dodecahedron #electricuniverse #fractalfield #fractality #geometriclife #gravityfalls #goldenratio #geometry #geometricmodels #hyperbolic #icosahedron #Lab333 #octahedron #Philostratus #platonicsolids #phi #RodrigoGrandArt #science #starshipc #tetrahedron #triangle #vitruvianman #vitruvius https://www.instagram.com/p/CQ3yJGSjTNf/?utm_medium=tumblr
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traslahuelladehegemon · 4 years ago
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— TITLE: The key differences that exist between male and female immune functions — TECHNIQUE: free will overexposure and underexposure of images DATE: 06/29/2021 SERIES: Abstinence DIMENSIONS: 3464 x 2309 pixels Autor: Rodrigo Granda _______________________ Photoediting quite loosely to capture the layers of colour and light 🙏🕋👁️ #ArtalDeAsens #awakening #alchemy #astronomy #amazing #awesome #contemplation #dodecahedron #electricuniverse #fractalfield #fractality #geometriclife #gravityfalls #goldenratio #geometry #geometricmodels #hyperbolic #icosahedron #octahedron #Philostratus #platonicsolids #phi #science #starshipc #tetrahedron #triangle #vitruvianman #vitruvius            (en Mexico City) https://www.instagram.com/p/CQtRJuIjlSp/?utm_medium=tumblr
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