杏吧原创

The jaws of infinity

Question: If I place two mirrors facing each other and stand between them, I
can see my reflection stretching away into what looks like infinity. Is this
really infinity? How small can my reflection get in this manner, and can it be
said to end in any particular way?

Answer: No mirror reflects 100 per cent of the light falling on it. If your
mirror is very good and reflects 99 per cent of the light, after some 70
reflections only 50 per cent of the light is left, after 140 reflections only 25
per cent of the light is left, and so on until none of the light remains to
travel between the two mirrors. In fact, most mirrors reflect some colours of
light much better than others and some colours are absorbed better by the glass,
so the multiple reflections that you see not only get darker but also more
colour-distorted as they recede towards infinity.

Even with perfect reflection of all colours you could never see infinite
reflections, for geometric reasons.

First, the faces of the two mirrors would need to be perfectly parallel. This
is actually impossible to achieve. There will always be a slight disparity in
their positions relative to one another. Hence the curving nature of such
reflections, until eventually the reflection is lost 鈥渁round the bend鈥�.

Secondly, even if the mirrors are perfectly reflective, perfectly parallel,
and really huge, your eyes are in the middle of your head, not on its edge.
Therefore, at some point, the receding and hence apparently more distant mirror
images would become smaller than, and hidden behind, the first reflected image
of your head. Even with a tiny camera and a giant pair of mirrors, the reducing
reflection size would eventually be smaller than the first reflected image of
the camera apparatus.

Hillary Shaw

Scunthorpe, North Lincolnshire

Answer: In theory, you could get an infinite number of reflections in the
mirrors, but only if they were perfectly parallel and you stood there forever.
Light travels at a finite speed (c) of roughly 3 脳 10⁸ metres per
second, so if your two mirrors were a distance of L metres apart and
you stood between them for t seconds you would be able to see
ct/L reflections. If your mirrors were 2 metres apart and you
stood between them for 1 minute, you should be able to see 9 billion
reflections.

However, in practice, you couldn鈥檛 set up the two mirrors to be perfectly
parallel. Let鈥檚 call the angle by which the mirrors deviate from being perfectly
parallel &dgr;&thgr;. The reflections will then become progressively further from the
centre of the mirror, until they fall off the edge. If &dgr;&thgr; is measured in
radians, and the mirrors each have a radius r, then the maximum number
of reflections you can see is r/L&dgr;&thgr;. If your mirrors each have
a radius of 1 metre, and are half a degree (0.0087 radians) away from being
perfectly parallel, you should be able to see 57 reflections.

Peter Bleackley

Astronomy Group

University of Leicester

Copper conundrum

Question: The Independent newspaper recently stated that 鈥渋f
whisking is something you do often, consider indulging in a copper bowl. The
copper reacts with egg whites to make them fluff up to a greater volume than
whites whisked in other bowls.鈥�

Is this true? If so, what is happening?

Answer: When you are whipping eggs, the kind of bowl you use does make a
difference.

On page 78 of On Food and Cooking: The science and lore of the
kitchen (Scribner, 1984) and also in Nature (vol 308, p 667)
Harold McGee reports that copper bowls produce a yellower, creamier foam that is
harder to overbeat than the foam you get from ceramic or stainless steel
bowls.

Copper ions migrate from the bowl and into the egg-white mix. These ions form
complexes with conalbumin, which is one of the proteins that is found in egg
whites. The conalbumin-copper complex is more stable than conalbumin on its own,
so it resists denaturing and unfolding. The complex is yellow, and visibly
colours the foam.

This extra stability affects the foam. When a copper bowl is not used,
whipping the foam unfolds or denatures egg proteins, which then collect and
coagulate at the air-water interfaces. The air bubbles in the foam are
stabilised by the coagulated proteins, and the whole foam stiffens. But if the
foam is overbeaten, the egg proteins become too denatured and coagulated, and
consequently they begin to clump. Once the foam reaches this stage and becomes
lumpy there is no saving it, as many cooks will have experienced to their
cost.

When the eggs are whipped in a copper bowl, copper ions bind to the
conalbumin fraction of the egg white and reduce the number of protein molecules
that are available to be denatured and coagulate. This makes the mix more stable
and therefore less subject to being overbeaten.

It is possible that copper does more than merely complex with conalbumin. It
may also react with sulphur-containing groups on other proteins and so interfere
with general coagulation as well.

Incidentally, conalbumin is very similar to human transferrin, which is an
iron-binding protein in our blood. Transferrin transports iron throughout our
body, and when the pH is above 6, all the transferrins can bind ions
like iron, zinc, manganese and copper. Conalbumin is so similar to transferrin
it was once renamed ovotransferrin, but the name didn鈥檛 stick.

Cream of tartar performs a similar function to a copper bowl and it has the
benefit of being much cheaper. But if you think that you can use a cheap iron
bowl instead of a copper one, you鈥檙e out of luck. Though iron forms complexes
with conalbumin, it doesn鈥檛 make a more stable foam. Zinc is not as good as
copper either.

It seems that the choice of copper bowls for whipping eggs is not a recent
discovery. The connection appears to have been known for more than 200 years,
and recipe books from as long ago as 1771 have mentioned the use of copper bowls
for beating eggs. One French recipe book from 1936 describes it in such an
offhand way as to suggest that it was common professional knowledge at the
time.

Joanne Nova

Canberra

This week鈥檚 question

Bottle battle: When I fill the kitchen sink to wash plastic baby bottles, the
bottles persistently float towards the stream of water, causing hot water to
splash around my kitchen as it hits the bottles. Why do the bottles do this? I鈥檇
have assumed they would drift away.

Austin Rugless

Balnarring, Victoria