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    The Visual Nerd In You Understands Curved Space
    By Mark Changizi | September 30th 2010 09:22 AM | 14 comments | Print | E-mail | Track Comments
    About Mark

    Mark Changizi is Director of Human Cognition at 2AI, and the author of The Vision Revolution (Benbella 2009) and Harnessed: How...

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    You’ve heard that space is curved – that’s gravity. You’ve also been told that you cannot really understand curved space. Sure, you can come to know curvy mathematics by studying general relativity or differential geometry, but you cannot grasp curved space in your bones…for the obvious reason that, in our everyday human-level world, space is flat, and so we have a brain for thinking flat.

    Or, at least, that’s what they say.


    But there is at least one variety of curvy mathematics that your brain comprehends so completely that you don’t even know you know it. It concerns your visual field, and your innate understanding of the directions from you to all the objects in your environment.


    In thinking about your visual field, it is best to imagine a sphere around your head, recording the directions to all objects in one’s environment. Call it the “projection sphere,” since it records in which directions objects project light toward us.


    So, if you are standing in front of a row of six vertical poles, then they will project onto your sphere as shown below. In this figure, one imagines that you, the observer, are at the center of the sphere, looking in the direction of the cross.




    Consider now the way these poles project…

    First, notice that each pole appears straight in your visual field. They are not straight in the figure above, but remember that the observer in the figure is at the center of the sphere looking out. Each pole is straight on this projection sphere -- and thus in your visual field -- because each is what is called a “great circle,” extending in this case from the bottom to the top of the sphere like lines of longitude.


    Second, observe that the poles are parallel to one another at the equator.


    Yet, despite being straight lines that are parallel to one another, they intersect! Namely, the lines intersect at the top and bottom of the sphere.


    Can this really be?


    It can really be, and it is possible because of the non-Euclidean nature of the geometry of the visual field.  
    The geometry that is appropriate for the visual field is the surface of a projection sphere, and the surface of a sphere is not flat / Euclidean, but, well, spherical.

    There are three main kinds of geometry for space: elliptical (including spherical), Euclidean (or flat), and hyperbolic.  
    How does one tell them apart? One way is to simply measure the sum of the angles in a square drawn in that space.

    In Euclidean geometry, the sum of the angles in a square is 360 degrees. But for elliptical geometry the sum adds up to more than 360 degrees. In hyperbolic geometries, on the other hand, the sum comes to less than 360 degrees.  Back to the visual field, then, let’s “draw” a square on it and sum up its angles.




    The figure above shows a square in your visual field. Why does it count as a square? Because (i) it has four sides, (ii) each side is a straight line (being part of a great circle), (iii) the lines are the same length, and (iv) the four angles are the same.

    Although it is a square, notice that each of its angles is larger than 90 degrees, and thus the square has a sum of angles greater than 360 degrees.  The visual field is therefore elliptical, and spherical in particular.

    One does not need to examine figures like those above to grasp this. If you are inside a rectangular room at this moment, look up at the ceiling. The ceiling projects toward you as a four-sided figure. Namely, you perceive its four edges to project as straight lines. Now, ask yourself what each of its projected angles is. Each of its angles projects toward you at greater than 90 degrees (a corner would only project as exactly 90 degrees if you stood directly under it).

    Thus, you are perceiving a figure with four straight sides, and where the sum of the angles is greater than 360 degrees.

    Your visual field conforms to an elliptical geometry!

    (The perception I am referring to is your perception of the projection, not your perception of the objective properties. That is, you will also perceive the ceiling to objectively, or distally, be a rectangle, each angle having 90 degrees. Your perception of the objective properties of the ceiling is Euclidean.)

    It is often said that non-Euclidean geometry, the kind needed to understand general relativity, is beyond our everyday experience, since we think of the world in a Euclidean manner. While we may think in a Euclidean manner for our perception of the objective lines and angles, our perception of projective properties -- i.e., the directions from us to the world around us -- is manifestly non-Euclidean, namely spherical.

    We do have tremendous experience with non-Euclidean geometry, it is just that we have not consciously noticed it. But once one consciously notices it, it is possible to pay more attention to it, and one then sees examples of non-Euclidean geometry at every glance.

    ~~~

    This piece was adapted from my book, The Brain from 25000 Feet (Kluwer). Mark Changizi is Professor of Human Cognition at 2AI, and the author of The Vision Revolution (Benbella Books) and the upcoming book Harnessed: How Language and Music Mimicked Nature and Transformed Ape to Man (Benbella Books).

    Comments

    Aitch
    Nicely explained, Mark, thanks Does that mean I don't have to do a degree to see the world through my eyes in curved space? .....I'm so glad I know that, I thought I was needing glasses....wink Aitch
    SynapticNulship
    I'm already well aware of curved space....

    Aitch
    Either my eyesight's improving, or you're looking more attractive, Sam.....wink
    Aitch
    Makes me wonder if Euclidean geometry is an artifact of flat drawing surfaces and early architecture. We can imagine for example a set of beams perpendicular to the ground, tied by struts which were measured on the ground. While the centre of gravity for the beams would cause them to splay ever so slightly relative to one another with respect to the core of the earth, the error margin for the accuracy of the struts would negate it and tell the ostensible story of true 90-degree perpendicularity.

    vongehr
    "that space is curved – that’s gravity"

    It is vitally the curvature involving the time direction that gives gravity, and this is, for the reason of the negative signature of the time component in the metric alone, absolutely, completely beyond imagination with help of the visual system. On top of this comes the mere four dimensionality, which is only naively "just one more". In four dimensions, curvature, rotation (no axis of rotation anymore!), boundary topology (no classification of topological surfaces in 4D anymore!) all are becoming fundamentally new (even without the added negative time metric factor). The visual system and its two dimensionality (retina is in a sense part of the brain), and the conformal symmetry only possible in 2D, means that we cannot grasp 4D space-time curvature at all.
    Hfarmer
    I don't know about that.   Such a simple visual understanding can help people who are not adept at differential equations (let alone solving coupled partial non linear differential equations such as the Einstein Field Equations).  
    Granted the intuition gained is not going to be 100% accurate and no replacement for the rigorous understanding that studying the math would give. 
    Science advances as much by mistakes as by plans.
    vongehr
    Hey, I gona tase you bro. My comment was not some badly veiled snobism about maths proficiency. I was after something Mark may be interested in, namely the impotence of using our visual system for insight into problems that are not even remotely there in three flat dimensions or curved 2D because of the fact that we use a two dimensional surface (retina) that does already a whole bunch of the pre-processing.
    As a kid, I tried staring long times, rotating mentally the projections and frame drawings of four dimensional bodies (e.g. hypercubes, hyper melon slices, ...) in order to at some point "see it". Why not - we see 3D, 4D is just one more - right? Now I understand: Two eyes give us 3D but three eyes would also give you only 3D. What you would need is a 3D retina in order to see 4D! And "see" means "think": The retina is basically an extension of the brain itself (see mylenation of visual nerve and embryo development) and directly maps also into the visual cortex for several different computational layers. It is not just our eyes, but the whole visual system is restricted to only grasp Gaussian 2D curvature in flat 3D space which is basically just totally flat in terms of the 4D space-time curvatures that lead to gravity.
     
    Bonny Bonobo alias Brat
    You could say that dolphins almost ‘see’ in 3D using echolocation.. They emit clicking sounds and listen for the return echo to determine the location and shape of nearby items in 3D. This combined with their eyesight means that they are able to see an object from 2 different perspectives simultaneously, therefore they must be one of the best candidates for being able to see things in 4D. They also havea very large brain to body ratio, larger than ours. Apparently blind humans have also been able to train themselves to echolocate. “Vision and hearing are close cousins in that they both can process reflected waves of energy. Vision processes light waves as they travel from their source, bounce off surfaces throughout the environment and enter the eyes. Similarly, the auditory system processes sound waves as they travel from their source, bounce off surfaces and enter the ears. Both systems can extract a great deal of information about the environment by interpreting the complex patterns of reflected energy that they receive. In the case of sound, these waves of reflected energy are called "echoes"”. See http://en.wikipedia.org/wiki/Human_echolocation,” Ben Underwood was diagnosed with retinal cancer at the age of two and had his eyes removed at the age of three and discovered echolocation at the age of five. “He was able to detect the location of objects by making frequent clicking noises with his tongue. This case was explained in 20/20:medical mysteries. He used it to accomplish such feats as running, basketball, rollerblading, playing foosball, and skateboarding..”
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Mark Changizi
    I certainly didn't mean to suggest that our visual system can handle all kinds of non-Euclidean geometry. Just one special variety, and a 2D one indeed. But it is often informally suggested that non-Euclidean geometry itself, the very notion, is fundamentally beyond what our brains can accommodate.
    Bonny Bonobo alias Brat
    Its certainly beyond what my brain can accomodate at the moment. Can someone please explain to me what the parallel postulate, also called Euclid's fifth postulate means when it states that.. ‘If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles”. I don’t understand the bit that says ‘if extended indefinitely, meet on that side on which the angles sum to less than two right angles.’ What side?
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    vongehr
    Of course I realize that you needed an attractive hanger to hang what you actually wanted to say from. However, I think the hanger is not so good. "You’ve heard that space is curved – that’s gravity. You’ve also been told that you cannot really understand curved space. ... Or, at least, that’s what they say." Not only is curved space not gravity, but also I wonder who says all this. "But it is often informally suggested that non-Euclidean geometry itself, the very notion, is fundamentally beyond what our brains can accommodate". Non-Euclidean geometry as in what ants living on a large rubber balloon universe would measure - who says it cannot be comprehended? What at least I say is that the very curvature that makes gravity is one that our brains have no way of imagining at all. By "curved space", "they" meant curvature of three dimensional space intrinsically (at least, or maybe even space-time curvature in 4D). Now, you could just not care about what you may feel is hairsplitting sophistry in unrelated fields that your readers also do not care about. I say, inside this hairsplitting, you may discover something interesting also for cognition, for example the reason of why we are stuck to flat 3D thinking (e.g. thinking that rotations are around an axis - ever wondered how a propeller plane in 4D knows where it is supposed to fly to?): The 2D retina surface structure and its mapping and remapping dominates the visual system. It is not just software and experience - it is the hardware.

    @ Helen: The statement is confusing. "if extended indefinitely" just makes sure that you know the two lines are not just short line segments like the first mentioned in "line segment intersects two straight lines". Just delete "if extended indefinitely", maybe it becomes clearer.
    Bonny Bonobo alias Brat
    Thanks Sascha, that now makes sense.
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Mark Changizi
    I think you overestimate our ability to really grasp even 2D non-Euclidean geometry.

    Non-Euclidean geometry -- even in 2D -- is so counter-intuitive that it took centuries for mathematicians to slowly come around to the conclusion that it may be possible at all, and not a contradiction. The parallel postulate seemed to be axiomatic, undeniable. 

    We now can be taught to think about non-Euclidean 2D geometry, but even here our intuitions are weak, for we end up imagining a 2D manifold embedded in 3-space, rather than really comprehending what the 2D non-Euclidean manifold is like. ..what it spatially feels like, so to speak, to be inside.

    My point is that the projection-sphere -- i.e., the space of directions from you out to the world -- is a 2D non-Euclidean space. I.e., the relevant space concerning how things project toward us is inherently a spherical one. (And our retinal surface at any time samples a portion of this space.) And our brains had to evolve to implicitly be experts at dealing with this spherical space. Our visual brains, in particular.

    Even so, our cognitive, or reflective, selves have little awareness of our visual system's spherical-space expertise, something I have found through nearly 15 years of discussion visual perception and illusions with scientists and laymen. And the illusion literature is rife with confusion about this, where the distinction between perceived angular size (its extent along a great circle of the projection sphere) and perceived distal size (its size, in meters, out in the world) is commonly not made. Both are often referred to ambiguously as "perceived size", and the discussion jumps back and forth between the two meanings, leading to incoherency. (This is something I have written about in my previous books, and all my illusion papers.)

    vongehr
    "I think you overestimate our ability to really grasp even 2D non-Euclidean geometry."

    Of course, I cannot even imagine a flat (i.e. Euclidean!) 2D torus, not even with embedding, actually not even a "truly" un-curved 1D torus (a ring). I do not quite know what you have against embeddings into a flat space of higher dimensionality. The fact that positive curvature restricts the topology of the space does argue for there being always an embedding (implied, maybe not "real"), but this is again hairsplitting maybe. Of course, without embedding, there is no way at all to imagine a curved space as such, let alone its global topology. We agree on all of this. My point is just that this is precisely due to our hardware restricting us to imagine flat 3D space and curvature is thus only imaginable as far as it is at most Gaussian and embedded in that flat 3D space.

    "rather than really comprehending what the 2D non-Euclidean manifold is like. ..what it spatially feels like, so to speak, to be inside" is not so difficult. What it feels or optically looks like can be imagined and even simulated; there are web sites showing what it looks like to fall into a black hole for example. What is difficult is: I cannot imagine how the 4D embedding looks like so that a sphere may have internally much more volume V than the sphere's surface area A = 4 Pi R2 suggests. However, I can imagine how it feels and looks like living with such a sphere and entering/exiting it. It plainly is as if all the objects that enter that sphere become smaller inside. We would get used to it and how it optically looks like quickly without even having to evolve different brains. There are probably people playing around in such spaces using virtual reality goggles as we speak.

    On that your actual point is about exactly the unconscious "expertise" and troubles with terminology in your field of research. Sure, that maybe should have been the hanger. But I know the problem - I just found out myself. Put anything to do with gravity and Einstein into the first few lines and people click on it like crazy. Put something important, say about how people are driven to produce ever more crap all over the sciences, and nobody gives a moist rat's behind. It is a silly little world we are living in, and the reason is not the projection sphere. ;-)