The Magic of Mirrors
(DRAFT: Liable to change)

Aaron Sloman
School of Computer Science

Alastair Wilson
http://alastairwilson.org/
Department of Philosophy

University of Birmingham

Installed: 9 Mar 2015
Last updated:
19 Apr 2015: Added: room corner reflected in two ways
12 Mar 2015: added picture of mirror on the floor
10 Mar 2015: pictures improved slightly, and some text improved.

This document is
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/magic-mirrors.html

A partial index of discussion notes is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html

CONTENTS

Myths and Mysteries about mirrors.

There are many online web sites and many publications since long before the world wide web existed, offering answers to the question:

Why do mirrors flip horizontally but not vertically?

An example is the answer sketched by Douglas Hofstadter in

The Mind's I: Fantasies and reflections on self and soul
(Edited by D. Hofstadter and D.C. Dennett)

Hofstadter wrote, in the "Reflections" section following Thomas Nagel's "What is it like to be a bat?" (Chapter 24):

There is a famous puzzle in mathematics and physics courses. It asks, "Why does a mirror reverse left and right, but not up and down?" It gives many people pause for thought, and if you don't want to be told the answer, skip the next two paragraphs. The answer hinges on what we consider a suitable way to project ourselves onto our mirror images. Our first reaction is that by walking forward a few steps and then spinning' around on our heels, we could step into the shoes of "that person" there in the mirror-forgetting that the heart, appendix, and so forth of "that person" are on the wrong side. On a gross anatomical level, that image is actually of a nonperson. Microscopically, the situation is even worse. The DNA molecules coil the wrong way, and the mirror-"person" could no more mate with a real person than could a nosrep!

But wait-you can get your heart to stay on the proper side if, instead, you flip yourself head over heels, as if swinging over a waist-high horizontal bar in front of you. Now your heart is on the same side as the mirror-person's heart-but your feet and head are in the wrong places, and your stomach, although at approximately the right height, is upsidedown. So it seems a mirror can be perceived as reversing up and down, provided you're willing to map yourself onto a creature whose feet are above its head. It all depends on the ways that you are willing to slip yourself onto another entity. You have a choice of twirling around a horizontal or a vertical bar, and getting the heart right but not the head and feet, or getting the head and feet right but not the heart. It's simply that, because of the external vertical symmetry of the human body, the vertical self-twirling yields a more plausible-seeming you-to-image mapping. But mirrors intrinsically don't care which way you interpret what they do. And in fact, all they really reverse is back and front!

http://themindi.blogspot.co.uk/
Go to Chapter 24 and search for "Reflections".

The well known psychologist and vision scientist Richard Gregory (who discovered and analysed many fascinating visual illusions, and founded the Bristol Exploratory, among other things), wrote a short review of The Mind's I in New Scientist December 1981, available online here.

In column 4 he criticises Hofstadter for not noticing that the left/right rotation of text in a mirror occurs because the text has been rotated about its vertical axis to face the mirror "giving left right reversal" whereas "twiddling" text around its horizontal axis gives up-down and not right-left reversal, adding "This has nothing whatever to do with mental mapping as Hofstadter claims it has on page 404". It is not clear whether he interpreted Hofstadter's analysis correctly.

A myth smashed on the floor

However, there is a very simple and obvious fact that appears not to have been noticed by hundreds (thousands?) of people who worry about, or publish papers about, the mirror puzzle. If, instead of thinking about real or imagined rotations of the viewer, or real or imagined head-stands by the viewer, you just put a mirror on the floor and stand next to it and look down, you'll find yourself flipped upside down in the mirror. Moreover the upside down person, will have front and back facing the same way as your front and back, unlike the person reflected in the original vertical mirror.
Mirror, mirror on the floor,
Not flipped front-back any more??

XX

The same happens if you attach a mirror to a ceiling, so that it is horizontal and facing down. There is a restaurant in Vienna (or was when A.S. last visited over 20 years ago) where the walls and some of the ceilings are covered in mirrors, including the ceiling of the men's toilet above the urinals, producing a somewhat disconcerting anti-gravity effect for anyone looking up!

So, to get the vertical flip, instead of rotating yourself, or imagining rotating yourself, just rotate the mirror through 90 degrees until it is horizontal, facing down from above, or facing up from below the viewer.

Note that if a mirror is not parallel to or perpendicular to a person's long axis, then the reflection of the person will not be parallel to the person.

However, in all cases Hofstadter's point about asymmetric body parts not being congruent with their mirror images is correct. E.g. in the reflection, your left hand is congruent with your right hand outside the reflection. That is independent of where the mirror is and where you are.

Ambulance Example

In the UK, and probably many other countries, an ambulance typically has the word "Ambulance", or local equivalent, displayed on the front in reverse, so that car drivers will be able to read it correctly in their rear-view mirrors.

Here's the word "Ambulance" with no mirror to mess it around.
XX

Notice that a vertical mirror will reverse the word "Ambulance" from left to right even if the mirror is not facing the word, as sketched below (with apologies for poor perspective drawing -- perhaps fixed later):

XX

However a horizontal mirror will not reverse the text left to right, but top to bottom, as depicted here:

XX

Two Mirrors

The comparison is perhaps clearest with both mirrors present simultaneously.
XX

Apologies for inaccurate geometrical relationships --
the reflections in the vertical mirror should be made to
tilt away to right, or the mirror should be made to appear
perpendicular to the wall, with the view-point changed.

And just to prove that it works with a real vertical mirror to one side:

XX

Or with two mirrors: one on the floor, and one on the side:

XX

Corner Example
(Added 19 Apr 2015)

The top-left square below is a photograph of a 3-D corner of a room, showing portions of the floor and two walls, with three different objects indicating three different (orthogonal) directions from the corner. The remaining squares have three more versions of the same picture, with left and right reflected, up and down reflected, and both left/right and up/down reflected, as indicated.

XX

Each of the four pictures could be an original picture of a 3-D configuration. Without the labels provided you could not tell which of the four is the original, since the objects shown could be stuck to the surfaces, and therefore information about gravity cannot be used to distinguish the pictures that have been reflected up/down.

NOTE: all the reflections are reversible. E.g. the top right scene if reflected horizontally becomes the top left scene. However, two scenes depicted in the same row or the same column are, like a left and right hand, incapable of being superimposed (even after rotations). They differ by one reflection. The scenes that differ by two reflections, e.g. top left and bottom right, or top right and bottom left, can be superimposed, like two left hands, or two right hands.

Youtube presentation on the mirror puzzle

There is a delightful and very popular Youtube video (Physics Girl, presented by Meg Chetwood), demonstrating what seems to be the theory proposed by Richard Gregory, and emphasising the fact that humans have left-right symmetry but not top bottom symmetry.
     https://www.youtube.com/watch?v=vBpxhfBlVLU

The demonstration is excellent except for the fact that left-right symmetry has nothing to do with what's going on, and neither does it matter how the thing to be reflected in the mirror is presented (right way up, upside down, etc.)

The actual transformation produced depends only on the orientation of the mirror, though how we think about it depends on how we describe ourselves, e.g. as having a top and a bottom, a left side and a right side, a front and a back.

One of the commentators referred to this image, which almost makes the point we have made:
http://upload.wikimedia.org/wikipedia/commons/f/fc/Cloud_Gate_boy_reflection.jpg

If the mirror is tilted at 45 degrees to a person's vertical axis it will reflect a vertical person as a horizontal one, neither parallel to nor perpendicular to the mirror.

What a mirror really does

Perhaps the most general way to think about what the mirror does is this: all the surfaces visible from the mirror's reflective surface form a large hollow container with a very complex shape. In producing the reflection, the mirror turns that container inside out, something like a sock being pulled inside out, except that a real sock may have different colours and textures on the inside and the outside, whereas the imaginary sock turned inside out by the mirror has exactly the same visible features on the inside as on the outside, which becomes the inside after the reversal! This explains why the reflection of a left hand is incongruent with the left hand, but congruent with the right hand. A left-hand glove turned inside out fits on a right hand, and vice versa. (This interpretation is due to Alastair Wilson.)

A familiar fact about psyche-d: a left hand glove will not fit on a right hand, and vice versa, as Kant noted. Wittgenstein wrote in Tractatus 6.36111

A right-hand glove could be put on the left hand, if it could be turned round in four-dimensional space.
But he does not seem to have noticed that even in 3-D space psyche-d can be turned inside out, in which case a left hand glove will become able to fit on a right hand, and vice versa.

It's possible that one of the commentators made these points in the many comments on the video, but we have not searched through all of them.

NOTE: Mirrors and Psyche-d.
Added 27 Mar 2015 The mirror problem was mentioned along with a collection of other examples on which misleading claims are frequently made, e.g. about binocular rivalry, and ambiguous figures, in this 'Usenet' discussion (in the Psyche-D group) in 1997.
   http://www.cs.bham.ac.uk/research/projects/cogaff/misc/binocular-rivalry

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Aaron Sloman
School of Computer Science
The University of Birmingham 




































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