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Ben Goldberger is an interesting origami artist from Israel whom I had the opportunity to meet at CDO Convention 2018. You should definitely check out his work, especially if you are interested in recursive, fractal-like models. One of his models which I really liked is this spider on a web, which combines geometric and figurative origami.
I used this model to test a sheet of paper which is sold by IKEA under the name Givande and which I got my hands on courtesy of Aleksander Biestek. It is sold as a gift-wrapping paper and it feels like a thin Kraft paper. It is black on one side and a natural brown on the other, though without the striping typical for Kraft paper. Precreasing, including reversing the creases, went well. Black paint did not come off the folded creases, as happens with some painted papers, and since good black papers are hard to come by, this alone makes this one worth evaluating. Thinness made it possible to fold quite precisely. This model does not challenge the paper’s strength too much, so it’s hard to tell if it would tear after lots of precreasing or under higher tension.
So, this paper has many good properties, but apparently it was not such a good choice for this particular model. The reason is that it does not hold the crease too well and the web in this model has some folds which are rather loosely locked. Therefore, for folding this particular model, I would now choose a thicker, more sturdy paper. Still, I think Givande has a lot of potential, and, for example, for a pure square twist tessellation, it would have been OK since in such a model all pleats are tightly locked and holding the crease is not as important a paper property. It may also be a good choice for many figurative models.
The Menger Cross or Jerusalem Cube is a 3D fractal created by recursively drilling cross-shaped holes into each face of a cube. Its surface area approaches infinity while its volume approaches zero!
Goldfinches photo made from repetitions of the same photo
Created with www.dumpr.net - fun with your photos
Millard Sheets Gallery, New Works Exhibition, Sept. 2004; 4 X 5, E100G film, lightprint; inspired by Frans Cuyck van Myerop's Trompe L'Oeil with Dead Birds, c. 1670-80
- ¿Beber de la sopa primordial?
- Sí, ¡pero solo con popote!
- Sí, ¡y de ambos lados del espejo!
- ¿Y del popote?
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Alternating between the Menger Sponge and its inverse at subsequent depths of recursion. Shown at depths 2,3,4 - cols; both starting conditions - rows. (motivated by this)
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Pentagonal remapping of a wristwatch. Another Escher/Droste style
remapping.
©2008 David C. Pearson, M.D.
"First Pick. O. L. Schwencke, lith., N.Y."
A cigar box label printed by O. L. Schwencke. Notice how a box featuring this label appears within the design of the label itself. This recursive picture-within-a-picture is called the Droste effect, which is named after the Droste cacao tin that featured an illustration of a nun holding the tin.
Building on his first book, On Intelligence, Jeff bravely presents a framework for how the brain works to produce intelligence from neurons organized into ~150 thousand cortical columns.
His decades of self-funded dedication to studying how the brain works affords a possibly unique and unifying perspective. In both books, though, he loses his way when speculating on the artificial brains of the future (with logical inconsistencies and overgeneralizations anchored on our biology). I think the first 112 pages are the best part of his new book. I’ll focus on that and save a brief critique of his AI constraints for the end.
In his first book, Hawkins presents a memory-prediction framework for intelligence. The neurons in the neocortex provide a vast amount of memory that learns a model of the world. These models continuously make low-level predictions in parallel across all of our senses. We only notice them when a prediction is incorrect. Higher in the hierarchy, we make predictions at higher levels of abstraction (the crux of intelligence, creativity and all that we consider being human), but the structures are fundamentally the same.
If that is not mind-bending enough, in his new book, Jeff extends the memory framework to the construct of “reference frames”. Everything we perceive is a constructed reality, a cortical consensus from competing internal models resident in many cortical columns, the amalgam of 1000 brains. Those models are updated by data streaming from the senses. But our reality resides in the models.
Here are the best parts of his new book, in my opinion. I revisit them to learn. Travelling without moving, as we’ll see…
“The cells in your head are reading these words. Think how remarkable that is.”
“If you ignore folds and creases, then the neocortex looks like one large sheet of cells, with no obvious divisions. The neocortex looks similar everywhere. Every part of the neocortex generates movement. In every region we have examined, scientists have found cells that project to some part of the old brain related to movement. The complex circuitry seen everywhere in the neocortex performs a sensory-motor task. There are no pure motor regions and no pure sensory regions.”
The cortex is relatively new development by evolutionary time scales. After a long period of simple reflexes and reptilian instincts, only mammals evolved a neocortex. “At some point millions of years ago, a new piece of the brain appears that we now call the neocortex. It starts small, but then grows larger, not by creating anything new, but by copying a basic circuit over and over. As the neocortex grows, it gets larger in area but not in thickness.” Given the recency, it’s “probably not enough time for multiple new complex capabilities to be discovered by evolution, but it’s plenty of time for evolution to make more copies of the same thing.”
• Vernon Mountcastle’s proposition from 1978: “All the things we associate with intelligence, which on the surface appear to be different, are, in reality, manifestations of the same underlying cortical algorithm. Darwin proposed that the diversity of life is due to one basic algorithm (evolution). Mountcastle proposed that the diversity of intelligence is due to one basic algorithm.”
Beyond the evolutionary time-scale argument, the brains’ vast flexibility to accept different, even prosthetic, sensory input changes and its ability to learn many different things point to a universal framework for learning.
• Cortical Columns are “the largest and most important piece of the puzzle.” They are roughly one square millimeter in size with 100K neurons. A mouse has one column per whisker. “Every cortical column is making predictions. We are not aware of the vast majority of these predictions unless the input to the brain does not match.”
• Learning through movement: “The brain learns its model of the world by observing how its inputs change over time. There isn’t another way to learn. Every time we take a step, move a limb, move our eyes, tilt our head, or utter a sound, the input from our sensors change. For example, our eyes make rapid movements, called saccades, about three times a second. With each saccade, our eyes fixate on a new point in the world and the information from the eyes to the brain changes completely.” We don’t perceive any of this because we are living in the model, which is predicting the next input to come, across all the senses. “Vision is an interactive process, dependent on movement. Only by moving can we learn a model of the object.”
“To avoid hallucinating, the brain needs to keep its predictions separate from reality. We are not aware of most of the predictions made by the brain unless an error occurs.”
“Thoughts and experiences are always the result of a set of neurons that are active at the same time (about 2% of the total). Individual neurons can participate in many different thoughts or experiences. Everything we know is stored in the connections between neurons. Every day, many of the synapses on an individual neuron will disappear and new ones will replace them. Thus, much of learning occurs by forming new connections between neurons that were not previously connected.”
Sequence memory (like predicting the next note in a melody or a common sequence of behaviors): “Sequence memory is also used for language. Recognizing a spoken work is like recognizing a short melody.”
• Locus of Predictions: “Oddly, less than 10% of the pyramidal cell’s synapses are in the proximal area. The other 90% are too far away to trigger a spike. For many years, no one knew what 90% of the synapses in the neocortex did. The big insight I had was that dendrite spikes are predictions. A dendrite spike occurs when a set of synapses close to each other on a distal dendrite get input at the same time, and it means the neuron had recognized a pattern of activity in some other neurons. When the pattern of activity is detected, it raises the voltage at the cell body, putting the cell into what we call a predictive state. The cell is primed to spike… and the cell spikes a little bit sooner than if it would have if the neuron was not in a predictive state.” And this inhibits other neurons from ever firing, the ones who were behind in that race. “When an input arrives that is unexpected, then neurons fire at once. If the input is predicted, then only the predictive-state neurons become active. This is a common observation about the neocortex: unexpected inputs cause a lot more activity than expected ones.” Predictions prime the pump, sub-threshold. “Predictions are not sent along a cell’s axon to other neurons, which explains why we are unaware of most of them.”
“Most predictions occur inside neurons. With thousands of distal synapses, each neuron can recognize hundreds of patterns that predict when the neuron should become active. Prediction is built into the fabric of the neocortex. As few as 20,000 neurons can learn thousands of complete sequences. The sequence memory continued to work even if 30% of the neurons died or the input was noisy.”
• Reference Frames: “The secret of the cortical column is reference frames. A reference frame is like an invisible, 3D-grid surrounding and attached to something” (like a map)
“Predicting the next input in a sequence and predicting the next input when we move are similar problems. Our sequence-memory circuit could make both types of predictions if the neurons were given an additional input that represented how the sensor was moving.”
“Most of the circuitry is there to create reference frames and track locations. The brain builds models of the world by associating sensory input with locations in reference frames. You need a reference frame to specify the relative position and structure of objects. Roboticists rely on them to plan the movements of a robot’s arm or body. Reference frames were the missing ingredient, the key to unraveling the mystery of the neocortex and to understanding intelligence. We showed that a single cortical column could learn the 3D shape of objects by sensing and moving and sensing and moving. Each cortical column must know the location of its input relative to the object being sensed. To do that, a cortical column requires a reference frame that is fixed to the object. The brain must have neurons whose activity represents the location of every object that we perceive.”
“Mammals have a powerful internal navigation system. There are neurons in the old part of our brain that are known to learn maps of the places we have visited” — the hippocampus and enthorhinal cortex, organs roughly the size of a finger.
“Place cells tell a rat where it is based on sensory inputs, but planning movement requires grid cells. Grid cells form a grid pattern. The two types of cells work together to create a complete model of the rat’s environment. Every time a rat enters an environment, the grid cells create a new reference frame to specify locations and plan movements.” In the new brain, these same cells and structures create models of objects instead of environments.
“Every cortical column learns models of complete objects. The columns do this using the same basic method that the old brain uses to learn models of environments. It is as if nature stripped down the hippocampus to a minimal form, made tens of thousands of copies, and arranged them side by side in cortical columns. That became the neocortex. Each patch of your skin and each patch of your retina has its own reference frame in the neocortex. Your five fingertips touching a cup are like five rats exploring a box.”
“Not all cortical columns are modeling objects. Language and other high-level cognitive abilities are, at some fundamental level, the same as seeing, touching, and hearing. The reference frames that are most useful for certain concepts have more than three dimensions.”
• Thinking is a form of movement: “The brain arranges all knowledge using reference frames, and thinking is a form of moving. Thinking occurs when we activate successive locations in reference frames.”
“A cortical column is just a mechanism that tries to discover and model the structure of whatever is causing its inputs to change” whether the structure of environments, physical objects or conceptual objects. “Reference frames are not an optional component of intelligence; they are the structure in which all information is stored in the brain. Every fact you know is paired with a location in a reference frame. Organizing knowledge this way makes the facts actionable” to “determine what actions are needed to achieve a goal.”
“To recall stored knowledge, we have to activate the appropriate locations in the appropriate reference frames. Thinking occurs when the neurons invoke location after location in a reference frame, bringing to mind what was stored in each location. The succession of thoughts we experience when thinking is analogous to the succession of sensations we experience when touching an object with a finger, or the succession of things we see when we walk about a town.”
• What and Where Pathways. “Your brain has two vision systems. If you follow the optic nerve as it travels from the eye to the neocortex, you will see that it leads to two parallel vision systems, called the ‘what’ visual pathway and the ‘where’ visual pathway.” If you disable one, you can identify what something is but not where, or vice versa. “Similar pathways also exist for other senses. There are what and where regions for seeing, touching, and hearing.”
“Cortical grid cells in What columns attach reference frames to objects. Cortical grid cells in Where columns attach reference frames to you body.” The distinction depends on where the inputs come from. “If a cortical column gets input from the body, such as the neurons that detect the joint angles of the limbs, it will automatically create a reference frame anchored to the body.”
“Your body is just another object in the world. However, unlike external objects, your body is always present. A significant portion of the neocortex — the Where regions — is dedicated to modeling your body and the space around your body.”
For abstract concepts like mathematics, there are difference reference frames one could use to learn. “Part of learning is discovering what is a good reference frame, including the number of dimensions.” History can be learned on a timeline, or geographically. “They lead to different ways of thinking about history. They might lead to different conclusions and different predictions. Becoming an expert in a field of study requires discovering a good framework to represent the associated data and facts. Discovering a useful reference frame is most difficult part of learning, even though most of the time we are not consciously aware of it. The correct reference frame to understand how the brain works is reference frames.” It's no surprise that the memory trick called the method of loci, or memory palace, is a good method for remembering a large sequential list of nouns.
From fMRI studies, “the process of storing items in a reference frame and recalling them via ‘movement’ is the same.”
“Nested structure and recursion are key attributes of language. Each cortical column has to be able to learn nested and recursive structure. Cortical columns create reference frames for every object they know. Reference frames are then populated with links to other reference frames. The brain models the world using reference frames that are populated with reference frames; it’s reference frames all the way down.”
• The Thousand Brains Theory of Intelligence: The prevailing view of the neocortex was a hierarchy of feature detectors, from edge detectors up to face detectors. Jeff argues that each and every column is a sensory-motor system. “When the eyes saccade from one fixation point to another, some of the neurons in the V1 and V2 visual regions do something remarkable. They seem to know what they will be seeing before the eyes have stopped moving. These neurons become active as if they can see new input, but the input hasn’t yet arrived. There are connections between low-level visual regions and low-level touch regions.” Mouse vision occurs in the V1 region; it does not depend on a hierarchy of vision abstractions.
“All cortical columns, even in low-level sensory regions, are capable of learning and recognizing complete objects. A column that senses only a small part of an object (e.g., from a patch of retina) can learn a model of the entire object by integrating its inputs over time.”
“Learning is not a separate process from sensing and acting. We learn continuously. When a neuron learns a new pattern, it forms new synapses on one dendrite branch. The new synapses don’t affect previously learned ones on other branches. Thus, learning doesn’t force the neuron to forget or modify something it learned earlier.” It’s additive.
“What a column learns is limited by its inputs. Columns in V1 can recognize letters and words in the smallest font. V1 and V2 learn models of objects, such as letters and words, but the models differ by scale.”
“Knowledge of something is distributed in thousands of columns, but these are a small subset of all the columns. This is why we call it the Thousand Brains Theory: knowledge of any particular item is distributed among thousands of complimentary models. The columns are not redundant, and each is a complete sensory-motor system.”
• The Solution to Sensor Fusion and the Binding Problem: “Columns vote. Your perception is the consensus the columns reach by voting.”
“If you touch something with only one finger, then you have to move it to recognize the object. But if you grasp the object with your entire hand, then you can usually recognize the object at once. In almost all cases, using five fingers will require less movement than using one.” (made me think of reading Braille with multiple fingers). “Voting works across sensory modalities (sight, touch, etc.)”
How? “Cells in some layers send axons long distances within the neocortex” between left and right-hand brain regions or between V1 and A1, the primary vision and auditory regions. “These cells with long-distance connections are voting. Cells that represent what object is being sensed can vote and will project broadly. Often a column will be uncertain, in which case its neurons will send multiple possibilities at the same time. Simultaneously, the column receives projections from other columns representing their guesses. The most common guesses suppress the least common ones until the entire network settles on one answer. The voting mechanism works well even if the long-distance axons connect to a small, randomly chosen subset of other columns”
• The Stability of Perception with ever-changing inputs: “What we perceive is based on the stable voting neurons. We are not consciously aware of the changing activity in each column.” Roughly 98% are silent at any given time and 2% are continuously firing. Consider the experience of an optical illusion duality (like the drawing of a pair of faces or vase); you can only see one at a time, and there is a delay if you force yourself to switch. “Recognizing an object in one sensory modality leads to predictions in other sensory modalities.”
• Attention: We have the perception of multiple objects in our visual field even though we can only attend to one at a time. “Attention plays an essential role in how the brain learns models. The brain can attend to smaller or larger parts of the visual field. Exactly how the brain does this is not well understood, but it involves a part of the brain called the thalamus, which is tightly connected to all areas of the neocortex. It is so intimately connected to the neocortex that I consider it an extension of the neocortex.”
• Consciousness: “Neurons form a continuous memory of both our thoughts and actions. It is this accessibility of the past — the ability to jump back in time and slide forward again to the present — that gives us our sense of presence and awareness. This is the core of what it means to be conscious. If we couldn’t replay our recent thoughts and experiences, then we would be unaware we are alive.”
“The neocortex does not directly control any muscles. The neocortex has to be attached to something that already has sensors and already has behaviors (the primitive brain). It does not create completely new behaviors; it learns how to string together existing ones in new and useful ways.”
“Instead of the neocortex using a hierarchy to assemble features into a recognized object, it uses hierarchy to assemble objects into more complex objects.” The assumption of hierarchy has been stumbling block for neuroscience for many decades.
“Reverse engineering the brain and understanding intelligence is the most important scientific quest humans will ever undertake. At one point I debated whether I should end right there. A framework for understanding the neocortex is certainly ambitious enough for one book.”
Yes, perhaps he should have. AI has been his failing. And, more abstractly, it is one of the grand challenges of biomimicry. While the brain provides the existence proof of an iterative algorithm compounding complexity and generating intelligence, it is a non-trivial exercise to capture the right level of abstraction when instantiating on a silicon substrate. Jeff seems to anchor on our biology to the point of having the wrong reference frame, so to speak, for his intuition. He asserts that certain aspects of our biology must be replicated in all artificial intelligences (e.g., a physically moving vision sensor vs a raster scan saccade) while dismissing countless other aspects of our biology, from ion channels to the goal setting regions of the brain.
We can logically see many idiosyncratic limitations in our biology, constrained by cellular sensors and compute, and need not replicate them in silicon. Similarly, there are limits in our silicon substrates (e.g., number of metal layers and lack of dynamic interconnect) that we need to address if synaptic fanout and long-range voting circuits are fundamental elements.
He is not alone is anchoring on the wrong elements of biomimicry. Neuromorphic spiking compute comes to mind. And this is why I did not even present his AI arguments, as they seemed so riddled with misguided leaps of intuition. You can see how he lost his way in his first book when he asserted that we could generate an AI, and then cut and paste key blocks of functionality like the ability to speak French from one AI to another.
Nevertheless, I am curious about his self-funded work on the brain, which might be a meaningful contribution on the biological side.
A hokusaiesque, recursive Peirce Quincuncial projection of this panorama of the skyscrapers in Tokyo's Shinjuku district.
This is the same model as Mélisande*'s "Stars Unlimited", David made an independent discovery.
Folded from a hexagon cut of a square of 30 cm on the side of Torreón paper.
Everything photographed and edited by me, no stock.
Our world is made of sides. I suppose that if beings from outer space would visit our planet, they would be extremely surprised about how divided we are, especially for such a small ball in the cosmos. Countries, regions, political parties, religions, education, various industries, wars, various laws, various rights: literally anything is a ground for division. The existence of these groups is so commonly accepted that many don't hesitate to actually define themselves through them: "I am an ecologist", "I am an atheist", "I am a democrat", and so on.
Yet, if one would step back for a minute, what looks obvious is how similar people are, despite those beliefs. Most feel the same coldness under the same rain, the same fear after the same noises during the same nights, the same illnesses after behaving the same way, the same love for the same relatives, the same mothers worrying about the same children, the same alcohol to forget the same things. Even when it comes to what we are physically made of, genetics studies show that Europeans, Americans, Africans, Japanese or even Siberians people all have common ancestors (that actually lived in Eastern Africa). In a nutshell, technically, we all are brothers and sisters.
From there, why this irrepressible need to create groups and subgroups? Why create categories? Are we that arrogant to need this illusion of being different? Do we feel better with boundaries around us? Is it that important to defend some kind of side, to the point it becomes vital to destroy other ones? In the light of our deep identity, how absurd is that?
The guy on the picture is obviously against propaganda, but by attacking it, he is also doing some kind of propaganda. This is why the picture is recursively shown inside the paper he is holding, in order to illustrate the paradox lying there: even if you are against propaganda, there is not much you can do, because fighting it is also doing some. I am helpless about it, of course, but still, I am wondering why we stuck ourselves in this foolish situation...
"... the fundamentally recursive character of observation -- namely, that stabbing is best witnessed, rather than experienced." - T.A. Wilson, excerpt from Plastic Tub Phrenologist (1920)
[hipstamatic portraiture - folded, spindled & mutilated]
Created with a remote shutter, composed with Photoshop and patience.
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Quick and dirty. Brought to you by packing tape (nice on those sensitive areas of the skin), a slightly crazy director, and S&M special issue.
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Updated pictures of an older (2013) design, see here.
Designed and folded by me in 2013, photo by Thomas Petri, edited by me.
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EZ-Star:
Paper: hexagon edge length ca. 10.5 cm, Kraft paper; dodecagon cutted from 24 cm Kami
Model: Evan Zodl
Book: Wonderous One Sheet Origami by Meenakshi Mukerji, p. 84-86
Recursive star:
Paper: Hexagon from DC Kraft wrapping paper
Model: Jorge Jaramillo
Book: Because Origami Vol.1 p. 7-10
When I folded Evan's star, it reminded me of Jorge's, so I searched for it, pursed the central star and compared them. They are very similar, Jorge's a bit smaller and with more precreasing visible, but a nicer locking and look from the back and side. I like both of them and it's fascinating to see, how differently the folding sequence is.
"Echoes of Observation," part of the 'Momentum' series, captures a silent symphony of artistic dialogue. Here, a spectator stands absorbed in twin reflections of reality—an introspective woman and a photographer in action—each echoing the live moment shared between the viewer and myself, the unseen observer behind the lens. The scene, framed in monochrome, juxtaposes the contemplative stillness of the subject against the active creation of art, alluding to the perpetual cycle where viewers and creators continuously influence and reflect one another. This photograph is a study in the recursive nature of observation, an intentional blend of life imitating art and art imitating life, where every moment is both a mirror and a window to a deeper narrative.
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Negative p1 yields concentric circles.
Another one of my entries for the Dyxum.com photo contest titled "Photos of Photos". I'm a digital photographer so even though I really do print occasionally, I also view on the iPad quite a lot ;-) I created this by takng a pic of Buzz, transferring it to the ipad by eye-fi then taking a pic of the iPad and so on.
Another twist tessellation based on one of Robert Fathauer's Fractal Tilings. Based on some preliminary tests, I think it might in theory be possible to make a recursive folding sequence for this, but it would be rather complicated and inelegant.
Something different. Like a tree lined avenue. A perfectly good shot that I photo shopped to oblivion with a feed back style process.
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The jester Time stalks darkly thro' the mead;
Beneath his tread contentment dies away.
Hearts that were light with causeless anguish bleed,
And restless souls proclaim his evil sway.
A poem by H.P. Lovecraft
My first test with a Droste/M.C. Escher-style conformal spiral.
©2008 D. C. Pearson, M.D.
[With thanks to these guys and to the MathMap script by Josh Sommers
and breic]
A print-quality still from a recursive particle-emitting system built in Processing
view the original size for better detail
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I keep applying to be an iStockPhoto contributor. They keep rejecting me. When I say "keep," I mean twice failed and hoping for third-time-charm. Maybe my work doesn't seem conceptual or original enough for them? Remembering my original recursive piano photo, which didn't have the best lighting or sharpness, I thought the concept might make my photography stand out, so I retook it with a tripod and proper lighting.
Bonus: unexpected pretty bokeh!
Finally Nugget gets the love of his life. Mrs Nugget suggested some minor design enhancements to their to be home for the season and our dear overjoyed Nugget started working on them right away.
He was so excited and busy working on his mistress's suggestions that he refused my recursive requests of posing for a family photograph.
I wish both of them a great future full of happiness.
Tk Cr Nugget, i shall see you soon...!!!
If you have missed Nugget's story, here it is