Do androids dream of electric sheep? Philip K. Dick
Fig. 4.1.Schematic representation of the principles of active electroloca- tion in fish. The electric organ (solid black bar) gives rise to a dipolar field pattern around the fish. The electric current follows the local field lines (solid lines with arrows). A nonconducting target object (circle) perturbs the flow of electric current, causing a local decrease in current density near the object. This decrease in current density translates into a de- crease in the transdermal voltage across the skin near the object. The spatial pattern of the transdermal voltage across the sensory surface rep- resents the electric image of the object.
The body surface is covered with thousands of sensors that measure local changes in voltage across the skin (transdermal potential). You can think of each sense organ as an electrode pair that measures the potential dif- ference across the highly resistive fish skin.
One interesting thing is that the field pattern and receptor distribution allow the fish to detect objects in all directions—an omnidirectional sens- ing capability.
Another observation is that the density of sense organs is rather high (several per square millimeter), suggesting that the system may be good for spatial localization of small prey targets that are just a few millime- ters in diameter.
Also, the sensor density tends to be higher near the mouth, suggesting that this is the region of the body where fine spatial localization might be most important.
(Redrawn and quoted from Nelson [205], with permission; the drawing originates in Heiligenberg [186].)
Let us return for a minute to our discussion of the electromag- netic imaging in fish which we started in Section 2.1, Page 22.
4.4 The line of sight and convexity 75
Which immediately intuitive mathematical concepts would become less intuitive if, instead of sight, humans used electric sensing of the kind used by Nile elephant fishGymnarchus niloticus? The cru- cial difference is that we would loose the concept ofline of sight, the archetypal straight line of our geometry. Indeed, electric sens- ing would allow us to “see”around objects; closer objects would not obscure the view of more distant ones.
Bill Lionheart aged 6 It is worth to remember that Euclid (or a later editor of Ele-
ments) defines a straight line as
a line that lies evenly with its points.
It makes sense to interpret this definition as meaning that a line is straight if it collapses to a point when we hold one end up to our eye. Therefore, a straight line is a line of sight!6
Chapter 10 contains a number of simple problems illustrating the relations between the concept of line of sight and that ofcon- vexity. Recall that a subsetXof then-dimensional Euclidean space Rn isconvex if it contains, with any pointsx, y ∈ X, the segment [x, y](Figure 4.2). HH HHH©©©© © © © © © © HH HHH r r @ @ @ @ x y convex set ¡¡ ¡ @ @ @ @@ r r x y non-convex set Fig. 4.2.Convex and non-convex sets.
Daina Taimina aged 12 The class of convex bodies can be characterized in terms of rela-
tion
“A(partially) obscuresB”.
Indeed, the set of convex bodies7inRn can be characterized as the maximal possible collectionCof bodies inRnsuch that
• C is closed under all rigid movements and similarity transfor- mations.
• If you look at two non-intersecting bodiesAandB, both taken from the collectionCandApartially obscuresBin your field of view, thenBcannot (partially) obscureA(therefore the relation “AobscuresB” isantisymmetricin the class of convex bodies). I leave the proof to the reader as an exercise.
We have already discussed the prominent role of order among basic “built-in” mathematical concepts of human mind. A strict or- der<is an anti-symmetric relation (that is, statementsx < yand y < xcannot hold simultaneously). The other part of the definition of strict order is that it is a transitive relation (that is,x < yand y < zimpliesx < z).
Now we have one more evidence of a special role of order in mathematical cognition. Indeed it appears that human’s visual pro- cessing system, when dealing with convex objects, frequently as- sumes that the “Aobscures B” relation is not just antisymmetric, but is also transitive (and hence a strict order): if a body A is in front of body B, and B is in front ofC, thenA is in front ofC. It is a systematic error of our brains, and it can be seen in many vi- sual paradoxes with non-existing objects. For example, have a look at Figure 4.3 and try to decide which parts of the contraption are closer to the viewer and which are more distant.
David Henderson aged 15
The chapters of geometry dealing with convex bodies contain a number of results which are surprisingly intuitive and self-evident. Here is one example:
A convex polytope (that is, a convex and bounded polyhe- dron) is the convex hull of its vertices (that is, the smallest convex set containing the vertices).
R. T. Rockafellar in his fundamental treatise on convex geom- etry emphasized the paradoxical status of this statement [394, p. 171]:
This classical result is an outstanding example of a fact which is completely obvious to geometric intuition, but which wields important algebraic content and is not trivial to prove.
In relation to convex bodies and their properties, our intuition about convex bodies is both very powerful and very misleading— perhaps because we are excessively confident in our judgement— and I refer the reader to Chapter 10 for more examples of both feasts and failures of our intuition.