Posts Tagged ‘vision revolution’

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), and first appeared so adapted at Sept 30, 2010, in Science 2.0.


Mark Changizi is Director 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).

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Daniel Piza of Brazil’s national newspaper, Estadão, listed my book, The Vision Revolution, among the top books of 2010. Lots of great company there, including Derek Bickerton, Richard Wrangham, and Michael Tomasello.


Mark Changizi is Director 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). He is working on his fourth book at the moment, tentatively titled Making Faces, about emotions and facial expressions.



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The hardback of The Vision Revolution has been out for one year, and I couldn’t be happier with the reaction it has received, including reviews in fantastic places like the Wall Street Journal and Sciam Mind and mentions in places like the New York Times. It even made New Scientist’s “best books of 2009” story! Soon it will appear in China, Korea and Germany.

There has, however, been one gnawing problem with the hardback.

…the problem is its hardbackiness.

To understand my trouble with hardbackiness, let me back up and explain what I was aiming for in writing the book.

As a start, let me first describe what I was not aiming for: Not an academic monograph, to be read only by specialists. Not a journalist-style coverage of a topic. And not a book about how to help your brain, like “20 ways to make your brain smarter than the Johnson’s next door.”

My aim was not only to write a book that is readable (and funny) to non-specialists (i.e., a “trade” or “popular” book). Rather, my aim was to build a book that is part of the scientific conversation.

By “part of the scientific conversation,” I mean that the book is filled with ideas and evidence that go beyond what is found in the technical journal articles.

That, I believe, is what makes a popular science book exciting to non-specialists and laymen: in reading the book they are not merely learning about science, but are witnessing a portion of the lively scientific exchange.

The reader is put within the scientific conversation itself.

I didn’t come to this philosophy about what makes a good popular science book on my own. As I struggled with the drafts of my first trade book proposals, I had the opportunity to meet with John Brockman (http://www.edge.org/3rd_culture/bios/brockman.html ), the noted literary agent, author and the founder of The Edge ( http://www.edge.org ). That’s his photo at the top. It was he who laid out this good-popular-science book philosophy to me, and although it sounded obvious after he said it, it by no means was obvious to me beforehand.

That’s what makes authors like Desmond Morris, Richard Dawkins, Stephen Pinker. Daniel Dennett and Andy Clark so compelling. It’s not merely that they write well, but that they’re making a scientific case for their viewpoint. …and you and I get to watch.

And so that’s what I did in The Vision Revolution, take the reader along as I lay out the case for a radical re-thinking of how we see. Color vision evolved for seeing skin and the underlying emotions, not for finding fruit. Forward-facing eyes evolved for seeing better in forests, not for seeing in depth. Illusions are due to our brain’s attempt to correct for the neural eye-to-brain delay, so as to “perceive the present.” And our ability to read is due to writing having culturally evolved to make written words look like natural objects, just what our illiterate visual system is competent at processing.

In aiming to be part of the lively scientific exchange, there was another thing I tried to inject into the book: I tried to not take things too seriously.

As I have discussed in an earlier piece [ http://www.science20.com/mark_changizi/mind_hacks_over_stacks_facts ], too often science is treated as a set of textbook facts. Textbooks usually give that impression, and even when they are careful to say that science is in fact deeply in flux, the textbook look and feel dupes most of us into imbuing the book with too much truthiness. This is especially a problem for the cognitive and brain sciences, because the object of study is the most complicated object in the known universe, and we very often don’t know what we’re talking about. (We don’t know jack: https://changizi.wordpress.com/2009/08/12/18/ )

And that brings me back to the most significant flaw with the hardback version of The Vision Revolution: its hardbackiness. The rigidity of a hardback suggests truthiness, and although I do believe the ideas I put forth and defend in the book are true, I don’t want the cover’s hardness to be part of my argument.

Luckily, The Vision Revolution is now out in paperback, and is so remarkably bendy that the reader cannot help but to read with that engaged maybe-this-is-not-correct mindset, rather than the oh-look-at-all-those-true-things-science-has-figured-out mindset. With that mindset, the reader will be in the right mindset to truly be “part of the scientific conversation.”


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).

This piece first appeared June 21, 2010, at Science 2.0. And, no, Brockman is not my agent.

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