An Incomplete Education - 3,684 Things You Should Have Learned but Probably Didn't

53* AN I N C O M P L E T E

Gene splicing is not to be co

genetic engineering mixed up

babies. In that case, an ovum

for success) are removed from

the father, and returned to the

lion in-vitro children in the wo

now

spe

an

I

wom

fert

cial

mat

wom

Y

Dr. James Watson, left, and Dr. Francis spe
Crick with their Nobel Prize-Twinning wom
model of the DNA molecule reim
laun

ing

But mightn't that lead to th

people who want to buy their w

the ethical problems of biote

around with Life Itself. Other

ing with characteristics like ph

and ultrawimps? What if a ge

or the test area? There are still

MANY ARE COL

Yes, the Big Chill is coming, b
underwear for another 3,000 t

Over the past billion years, t
which ice built up at its poles,
ages," when glaciers advanced
punctuated by 10,000-year "in
the vast ice sheets. We live at t
ice age wound down about 7,0
encased much of North Ameri
grees colder than they are now,

E EDUCATION

onfused with cloning (see page 540). And don't get
with in-vitro fertilization, the creation of test-tube
(more likely, several ova, to maximize the chances
the mother, fertilized in a petri dish by sperm from
e mother's womb. Today there are more than a mil­
orld; in fact, the first, Louise Brown of England, is
w all grown up. And the term is unfair: The kid
ends only a couple of days in a dish, as an egg V25 of
inch in diameter.
In a newer procedure, so called prenatal adoption, a
man needn't even produce her own egg. Rather, the
tilized ova of another woman, who has been artifi­
lly inseminated with sperm from the first woman's
te, can be implanted in the womb of the first
man and the resulting fetus carried to term.
Yet another technique allows the freezing of eggs,
erm, and full-fledged embryos for future use. A
man might deposit a fertilized egg, meant to be
mplanted later, once she's sown her wild oats or
nched her career. Or she could rent a womb, pay­
someone else to carry her child.
he creation of a class of drone-moms, employed by
way out of morning sickness? Ah, that's just one of
echnology, the blanket term for all this fiddling
r sticklers: What if the wrong people start tamper­
hysique, building a brave new world of superjocks
enetically engineered microbe escapes from the lab
l some eensy bugs to be worked out.

LD BUT FEW ARE FROZEN

but you won't need your industrial-strength thermal
to 20,000 years.
the earth has experienced three long periods during
, each period made up of several 100,000-year "ice
d to cover much of the world. These ice ages were
nterglacials," warm spells marked by the melting of
the end of such a temperate time-out; the last great
000 years ago. At its peak, 20,000 years ago, glaciers
ica, Europe, and Asia. Days were about eleven de­
, forcing humans and animals southward.

It's not hard to see how an ice age is caused b
summers cool enough that the previous winter's
sons' snows accumulate and compact to form glac
thermostat? The cold facts have been hotly debat
accepted—the "astronomical" theory—states that
earth's position relative to the sun seem to have la
the amount of solar radiation the earth receives.

Because of the gravitational pull of the sun and
wobbles on its axis like a toy top slowing down. E
scribes a circle in space. The axis also tilts, causing
Pole tips away from the sun, it's winter in the Nor
angle of tilt is 23V2°, but every 41,000 years it m
again. Perhaps the most important cycle is a cha
orbit—from nearly circular to highly elliptical
100,000 years due to the gravitational tug of fellow
of these three cycles is to place the earth farther
times, cooling the planet into an ice age.

If the astronomical theory makes sense to mo
them get all worked up about an imminent ice age
thrown up by man-made pollution and increased
thicker cloud cover generated by jet vapor,
heat we get from the sun. (As so often hap
science, other people worry that some of these ver
earth too hot, not too cold.) Even if pollution and
dust to cause a slight drop in temperature (and
sweaters a lot in the summer of 1991, right after M
Philippines) most scientists agree that this will no
of the cosmos. It'll be thousands of years before th
a metaphor. In the meantime, ours may ultimately
history—dangerously warm, thanks to the greenh

SOME LIKE IT

Leave your car parked all day in the sun with the
return to an interior much hotter than the air ou
glass admits sunlight but traps heat. Carbon diox
has a similar physical property; it is permeable to
opaque to infrared radiation (heat). We need so
along with water vapor and ozone, an unstable
long as C 0 2 stays in its current proportions, i.e.,
total atmosphere.

SCIENCE 533

by a temperature drop, creating
snow never melts. Several sea­
ciers. But what turns down the
ed, but the theory most widely
t three periodic changes in the
aunched ice ages by influencing

moon on the equator, the earth
Every 22,000 years or so, it de­
g the seasons. When the North
rthern Hemisphere. Today, the
oves from 22° to 24° and back
ange in the shape of the earth's
l and back to circular—every
w planets. T h e combined effect

away from the sun at certain

ost scientists, why do some of
e? They insist that a veil of dust
volcanic activity, together with

will lessen the amount of
pens in the wacky world of
ry same problems will make the
d volcanoes do kick up enough
we did find ourselves wearing
Mount Pinatubo erupted in the
ot alter the primordial rhythms
he ice age gathers steam, to mix
y be the warmest interglacial in
house effect, up next.

HOT

e windows rolled up and you'll
utside. That's because window
xide ( C 0 2 ) , the gas we exhale,
o solar radiation (sunlight) but
ome C 0 2 to act as insulation,
form of oxygen. N o sweat, as
, only about .03 percent of the

534 AN I N C O M P L E T E

In the last 100 years, howev
about 15 percent, thanks to th
spread deforestation, which h
sized thousands of tons of C
in the atmosphere, the hotter
2°C to 4°C ( 3 . 6 T to 7.2°F), w
think could happen by this cen
and submerge coastal cities;
floods; and decrease wind circ
pleting the fish supply. Alrea
than they were a century ago
taken place in the last two dec

As is usual in the apocalyp
a boon, not a bane. First of
C 0 2 stimulates photosynthesi
tivity could soar. Others thin
period (see preceding entry)
however, side with the Britis
time ago that the release of C
may be the most important e

Don't confuse the greenho
such man-made chemical co
used as spray-can propellants
in the United States and much
will destroy the ozone layer (
the planet from deadly ultravi
skin cancer and cataracts. But
mysterious hole in the ozone
peared every winter since, is a
It could be a transient, fluky w
cal event—or the awful thing
Aquanet.

ONE SINGS, TH

As the host said, rolling his ey
stay together. She's so left hem
not that one of them's from P
their personalities epitomize t
to deal with the world. T h e le
body, is responsible for langua

E EDUCATION

ver, the amount of atmospheric C 0 2 has increased
he burning of coal, oil, and natural gas and to wide­
as eliminated plants that would have photosynthe-
0 2 into oxygen and carbohydrates. The more C 0 2

we get. A rise in the global temperature of a mere
which some climatologists and a brace of computers
ntury, would melt polar ice caps, causing seas to rise

upset weather patterns, triggering droughts and
culation, playing havoc with ocean currents and de­
ady global average temperatures are 1.1 °F warmer
o, and virtually all the hottest years on record have
cades.
pse biz, a few people see the greenhouse effect as
all, they insist, it won't get all that hot, and since
is and increases plant yields, agricultural produc­
k the warming trend could extend our interglacial
and even stave off the next ice age. More people,
sh scientific journal Nature, which observed some
C 0 2 from all those fossil fuels we've been burning
environmental issue in the world today.
ouse effect with deodorant doom, the notion that
ompounds as jet exhaust and chlorofluorocarbons
and in air-conditioning (and now being phased out
h of Europe but not in most of the rest of the world)
(ten miles up and twenty miles thick) that protects
iolet and other high-energy radiation and you from
feel free to fret. In 1985, scientists reported a large,
layer over the South Pole. The hole, which has ap­
as big as the continental United States and growing.
weather phenomenon, a previously unnoticed cycli­
g everyone said would happen if we overdid the

HE OTHER DOESN'T

yes at the departing couple, "I don't know how they
misphere, and he's so right hemisphere." He meant
Peshawar and the other's from Pittsburgh, but that
the different ways the two halves of our brain seem
eft hemisphere, which controls the right half of the
age and logic; the right hemisphere, which controls

the left half of the body, handles such intuitive, no
and spatial relationships.

Uncovering this dichotomy earned a 1981 Nob
Caltech psychobiologist who has experimented wi
had stopped speaking to each other. Ordinarily, th
connected by a bundle of millions of nerve fibers,
lows signals to pass between the hemispheres and e
tegrated unit. By observing epileptics whose corpo
prevent the spread of seizures, Sperry learned how
chores. In one classic experiment, he showed a di
sphere simultaneously. (What the right eye sees i
sphere, and vice versa.) The left, verbal hemisphe
knife and the right, nonverbal hemisphere was show
to name what he saw (to use language, a left hemi
knife. Asked to feel about with his left hand (a sp
the right hemisphere) and pick up what was in th
spoon from a group of objects. The right brain di
was doing.

Until recently, the division of labor between left
cut and accounted for a variety of functions: intell
versus abstract thinking, objective versus subjective
holistic perception. But now the head honchos the
of issues. Most agree that in 95 percent of right-ha
handers, the left hemisphere is specialized for lang
gests that the right brain handles some imp
recognizing narrative and humor (storyline and pun
terpreting tone of voice, making metaphors. Sc
thought that the rational left brain was definitely
quishing control to the right only during sleep
Now it appears that there may be complementary
Each hemisphere has a specialty and kicks in when

The specialization may have less to do with wha
than with how it's done. The right hemisphere se
creative generalist, trying lots of solutions until one
left brain seems to be a plodding specialist, good
that are already familiar. Physical differences b
hemispheres support this hypothesis: The right hem
long fibers plugging into many different areas of th
can schmooze and tune into the grapevine to solv
The left brain contains shorter fibers tapping a sm
allowing it to do more detailed work. Still more s
these neurologists ever go to the movies?) show th

SCIENCE 535

onverbal processes as emotions

el Prize for Roger Sperry, the
ith people whose hemispheres
he two halves of the brain are

the corpus callosum, that al­
enables us to function as an in­
ora callosa had been severed to

the hemispheres divvy up the
ifferent picture to each hemi­
is processed in the left hemi­
ere was shown a picture of a
wn a picture of a spoon. Asked
isphere skill), the subject said
piral spatial skill, controlled by
he picture, the subject chose a
idn't know what the left brain

and right seemed fairly clear-
lect versus intuition, concrete
e understanding, linear versus
emselves are split on a number
anders and 67 percent of left­
guage. But new evidence sug­
portant language functions:
nchline), in­
cientists also
y boss, relin­
and dreams.

dominance:
n duty calls.
at the task is
eems to be a
e works. The

at problems
between the
misphere has
e brain, so it
e a problem.
maller space,
tudies (don't
hat the right

AN I N C O M P L E T E

brain handles novelty, the lef
tered with the right hemisphe
telegraphers interpret Morse
the right. Nonmusicians reco
sicians use the left.

One interesting bit of conje
déjà vu—the feeling that wh
past—may be nothing more t
reach one hemisphere a split
event is processed twice.

ST

In the last thirty years, resear
pens when a foreign invader (
First, the army—our trillion
alerted. These cells are of tw
stantly patrol the body looki
commandos called T and B
where T cells mature, B for b

The soldiers in this army
(Yes, of course, the military m
doesn't cut it.) At the first sig
begin to swallow up the enem
tiny piece called the antigen, w
to attract helper T cells traine
apparently has on hand a T ce
lions of antigens.) The macrop
stimulates the helper T cells to r
which causes the creation of st
it is to clobber body cells alrea
release a lymphokine that orde
nodes. (Gives a whole new me

Helper T cells then produc
to stop reproducing and start
cific antigens. T h e B cells are
which either fight the invader
vour them. The helper Ts,
phokine that boosts T cell act
helps macrophages digest the

When the invaders are van

E EDUCATION

ft brain what's old hat. Unfamiliar faces are regis­
ere, familiar faces recognized with the left. Seasoned
e code with the left brain, young whippersnappers,
ognize melodies with the right hemisphere, but mu­

ecture to come from split-brain theory: the idea that
hat's happening now has already happened in the
than a neurological glitch that causes information to

second before it reaches the other, so that a single

ATE OF SIEGE

rchers have pieced together a picture of what hap­
(e.g., a virus, bacterium, or parasite) enters the body.
n white blood cells, born in the bone marrow—is
wo major kinds: phagocytes ("cell eaters") that con­
ing for intruders to devour, and lymphocytes, little

cells ( T is for thymus, a small gland in the neck
bone marrow, the birthplace of B cells).

communicate via special proteins called cytokines.
metaphor is tired, but "the body's bowling team" just
gn of attack, special phagocytes called macrophages
my, taking from each the equivalent of its dog tag—a
which a macrophage then displays on its own surface
ed to recognize that particular antigen. (The thymus
ell able to recognize each of nature's hundreds of mil­
phage secretes a cytokine called interleukin-1, which
reproduce. The helper Ts in turn secrete interleukin-2,
till more helper T cells and of ki/krT cells, whose job
ady taken over by the invader. The new helper T cells
ers the production of B cells in the spleen and lymph
eaning to the term "sick day," doesn't it?)
ce yet another lymphokine that tells B cells it's time
t making antibodies, proteins tailored to fight spe­
e capable of creating millions of different antibodies,
rs or flag them so phagocytes can recognize and de­

bless 'em, also secrete gamma interferon, a lym­
tivation, assists B cells in producing antibodies, and
e enemy.
nquished, suppressor T cells tell the rest of the im-

mune system to call it quits. Turning off the immu
launching it; some forms of blood cancer seem to
gone wild. And the immune system can mistakenl
as in such autoimmune diseases as rheumatoid
erythematosus. (Allergies are basically an immune-
less invaders like dust and pollen.)

After the battle's over, phagocytes clean up the d
tein fragments. Certain B and T cells, called mem
and the lymphatic system, on guard against renewe
Vaccination is a sort of basic training for memor
weakened disease-causing substances into the bo
that they'll recognize the invader should it ever app

The good guys don't always win, of course. Cold
stantly mutating to escape detection, and chemic
macrophages, which can't digest them. But for eve
suffer, thousands of sneak attacks have been thwar

There's one attacker against which the body see
virus that causes A I D S , enters the body hidden
macrophages in blood, semen, or vaginal fluid from
apparendy hijacks the victims helper T cells when th
trusion. It keeps more or less quiet for months or
until they begin to divide, perhaps in response to so
only do the helper T cells fail to sound the alarm th
B cells, but they are themselves spreading the dise
the form of opportunistic infections, meet no oppo
the kill.

There is, however, good news on other fronts. R
arsenal of the immune system to fight cancer. Cro
they've cloned the resulting hybrid to produce antib
incipient tumors. These so called monoclonal anti
tinely used to deliver drugs or radiation to a specific
passing healthy tissue.

Perhaps the most dramatic advances come from
psychoneuroimmunology, the study of interaction
nervous, endocrine, and immune systems. Scienti
that a body under stress produces an excess of
macrophages coping with all that Cortisol can't see
we catch more colds when we're stressed out. We
stimulate the brain to produce natural painkil
enkephalins, and that it might even affect T cell
do the brain and immune system communicate?
may speak to the white blood cells through the cy

SCIENCE H7

ne response is as important as
be the result of T and B cells
ly attack the body's own cells,
arthritis and systemic lupus
-system overreaction to harm­

debris—dead cells, spilled pro­
ory cells, remain in the blood
ed attack by the same antigen.
ry cells; it introduces dead or
dy, priming memory cells so
pear again in full force.
d viruses, for example, are con­
cals like asbestos overwhelm
ry bout with flu or herpes you
rted.
ems to be powerless. HIV, the
n inside helper T cells and
m an infected person. The virus
hey come to investigate the in­
years in these hostage T cells
ome subsequent infection. Not
hat would activate killer T and
ease. Second-wave invaders, in
sition when they march in for

Researchers are harnessing the
ossing spleen and cancer cells,
bodies that scout the body for
ibodies may someday be rou­
c diseased site in the body, by­

m the relatively new field of
s between the mind and the
ists have found, for example,
the steroid Cortisol. Because
em to handle other infections,

know that exercise seems to
lers, called endorphins and
production. Why? And how
It looks as though the brain
ytokines, the protein messen-

53» AN I N C O M P L E T E

gers—and the white blood ce
mately be found to work not
back loop. Is it the body's d
new metaphors.

G

O K , here's a quiz: The Hum
funded science project that be
press conference hosted by the
ter Tony Blair; (b) a vast gove
and ended in April 2003, whe
had closed many gaps in the p
science project that began in 1
Nature published what it ca
genome; or (d) all of the abov

You guessed it: d. And tha
Genome Project. Its mission,
beholder.

But let's back up. You've re
already (see page 530) and h
crammed along a tightly coile
one hundred trillion cells that
(but not red blood cells, whic
information contained in that
volumes or, if you prefer, 125 M
velopment, growth, and death
gram, it's relentlessly binary: T
railings of the famous spiral st
for making proteins, the basic

The Human Genome Proj
U.S. Department of Energy a
tify the thousands of human g
lion base pairs (or nucleotides
97 percent of the nucleotides
tein coding.) It was a complic
For one thing, as scientists go
that the number of genes hum
mates: from upward of a hund
less) final count of twenty-fiv

E EDUCATION

ells may talk back. The immune system may ulti­
just as an army, but as part of an elaborate feed­

discussion group? Its town meeting? Stand by for

G E N E S "R" US

man Genome Project was (a) a vast government-
egan in 1990 and ended in 2000 with a fancy joint
en-president Bill Clinton and British prime minis­
ernment-funded science project that began in 1990
en the National Institutes of Health decided that it
previous calculations; (c) a vast government-funded
1990 and ended in October 2004 when the journal
alled the most complete sequence of the human
ve.
at's the beauty—and complexity—of the Human
, essence, and conclusion are all in the eyes of the

ead, or at least we've talked, about genes and D N A
how there are thousands of genes in each of us,
ed strand of D N A in the nucleus of every one of the

make up the human body, including hair and nails
ch have no nucleus). And as we've mentioned, the
t strand equals the data in a thousand encyclopedia
Manhattan phone books. It programs the birth, de­
h of each of us. And like any good computer pro­
Three billion pairs of chemical code letters form the
taircase, a.k.a. the double helix, and are responsible
c building blocks of life.
ject ( H G P ) was originally set up in 1990 by the
and the U.S. National Institutes of Health to iden­
genes and map the exact sequence of the three bil­
s) that make them up. (It's worth noting that about

are "junk"—that is, they don't correspond to pro­
ated job, and in some ways a humbling experience.
ot closer to completing the sequence, they realized
mans have is considerably lower than original esti­
dred thousand back in the mid-1990s to a (more or
e thousand, which puts us on par with the puffer

fish and the mustard weed. As Dr. Francis Collin
"If we wanted to claim our special attributes on t
count, we've taken a serious hit." (Since this disc
plain what makes humans different from other l
cusing on the aforementioned "junk" parts of the
might turn out to be some kind of complicated g
mean that human complexity arises not from th
selves but from the way those proteins are wired t

For a government agency, the H G P did a pretty
of the aforementioned dates count as the completi
ule (the project had been slated to take at least fif
your accounting system, probably under budget.

The mapping of the human genome is seen as
mation on, among other things, four thousand gen
retically, identifying the specific gene for a diseas
which in turn could lead to cures for everything fro
to A L S (Lou Gehrig's disease) to severe combin
called boy-in-the-bubble syndrome (all of which,
been identified, thanks to the H G P ) . It could also
ing treatment plans for particular individuals. Acco
day your personal genome might be interpreted a
to a smart card. Then, when you got sick, you co
hand over the card; doctors would be able to see all
take it personally, we all have them), identify prec
prescribe just the right treatment for your specific

But as you may have guessed, we're still a long
there have been major advances in the number of
about 150 in 1990 to about 1,500 in 2002), and ev
cer research (in 2004, scientists located a region o
risk of getting lung cancer). But once you've dis
next steps—understanding the pathogenesis of t
eventually, preventing it—are difficult, time-cons

There are also ethical complications, which tend
the science evolves. D o you really want to be in o
biological fate? What would happen to unborn b
the gene for, say, Huntington's chorea or cystic fib
cide to abort? What if you were told that your fetus
chance of developing breast cancer or Alzheimer's
lot of good years to decide to cut off in the name of
are legions of parents who wouldn't be happy to h
gene for obesity—or homosexuality.

Granted, we've calmed down about some of the

SCIENCE 539

ns, director of the H G P , put it:
his planet come from the gene
covery, scientists hoping to ex­
iving creatures have begun fo­
genome, which, many believe,
genetic circuitry. If so, it could
e basic building blocks them­
together.)
y good job. Regardless of which
ion, it came in ahead of sched­
fteen years) and, depending on

the first step in tracking infor­
netically linked diseases. Theo­
se could lead to new therapies,
om Lyme disease to Alzheimer's
ned immunodeficiency—the so

and many others, have in fact
o give doctors a leg up in devis­
ording to its proponents, some­
nd the information transferred
ould walk into any hospital and
l the glitches in your code (don't
cisely what the problem is, and
genetic makeup.
way from such a scenario. Yes,

disease genes identified (from
ven in their application to can­
f genes that increase a person's
scovered the elusive gene, the
the disease, then treating, and
suming, and costly.
d to multiply, not disappear, as
on that biggest of secrets, your
babies who were found to have
brosis? Would their parents de­
s had a better-than-50-percent
in middle age? That's an awful
f a mere probability. And there
hear that their child carried the

se issues. For one thing, scien-

540 AN I N C O M P L E T E

tists have explained that it's ra
they believe it has to do with
that genome "junk" works—a
with Judy Garland. Meanwhile
selves to the proteins they mak
field mouse, but our superior p
Stuart Little and Mickey, not
ward the conclusion that it's n
rather some of both. After all
to mention the dearth or glut
sity as the fat gene.

Nowadays, the ethical debat
dane questions; for instance, h
being used against us by emplo
for them to obtain, given how
the hair on the barbershop floor
Nondiscrimination Act throug

On a more philosophical le
clear: Genetically speaking, p
Scientists now say that 99.9 p
tical. Not only that, but 50 pe
we're even closer to the genet
about that the next time you r

CLONING AND

It all started with a little lamb
the world's attention as the fir
of another. Until then, most s
was possible. But Dolly, name
whole new scientific realm—a

But we should back up h
"cloning" is actually a pretty
originally described an exact
produced, such as when a straw
propriate distance from the p
1952 when biologists made a
testinal epithelial cell of a tadp
two decades scientists had c
clones were cultured from emb
since she was cloned from a

E EDUCATION

arely a single gene that makes Jimmy gay. Instead,
how numerous genes interact with each other, how
and how closely the prepubescent Jimmy identifies
e, the focus has been shifting from the genes them­
ke. (We may have the same number of genes as a
protein production means we're the ones exploiting
vice versa.) In any case, current thinking leans to­
neither nature nor nurture that decides our fate, but
l, we know that cultural and economic factors—not
of fast-food chains—have as much to do with obe­

te over genetic research revolves around more mun­
how will we keep personal genetic information from
oyers and insurance companies? (It wouldn't be hard
it's all right there, in blood, fingernail clippings, and
r.) Some U . S . politicians have tried to push a Genetic
gh Congress, but so far it has failed to gain passage.
evel, all of this genetic scrutiny has made one thing
people are a lot more alike than they are different.
percent of human beings' D N A sequencing is iden­
ercent of our genes are the same as a banana's. And
tic makeup of a fruit fly. You might want to think
reach for the flyswatter.

THE STEM CELL DEBATE

named Dolly. Born on July 5,1996, the ewe caught
rst mammal to be cloned from the adult body cells
scientists didn't believe that cloning from adult cells
ed after country singer Dolly Parton, opened up a
and a big can of worms.
here. For all its cutting-edge, sci-fi associations,
old term. Coined by a horticulturalist in 1903, it
genetic copy of an individual organism, asexually
wberry plant sends out clones to take root at an ap­
arent plant. Cloning took on new connotations in
new frog from the D N A in the nucleus of an in­
pole. After that, the floodgates opened, and within
cloned mice, pigs, sheep, and cattle. All of these
bryo cells. Dolly, however, was in a class by herself,
an adult cell—and her birth decimated the then-

generally accepted theory that once cells reached a
totipotent (that is, able to develop into a complete

Somatic cell nuclear transfer—the fancy name fo
least in theory. A cell is taken from an adult anima
D N A inside it, is pulled out and placed beside an em
(Think of it as sort of a blind date in a petri dish.) Th
nudged together by a mild electrical current and bat
potion. The egg is essentially duped into thinking it
best-case scenario it starts dividing like crazy. Altho
being a mad-scientist experiment exclusively create
only the very beginnings of the process actually hap
ation, the blastocyst (the ball of dividing cells that
transferred to the womb of a surrogate mom. (Dol
anyway—were only "grown" in the lab for seven day
a blackface ewe s womb. The surrogate then carried

As simple as all this sounds, cloning is notorious
clone that survives beyond birth, there have been,
as many as a thousand unsuccessful ones. Obvious
There are also questions about whether clones
Many cloned animals were born ill or deformed, o
includes, alas, our friend Dolly, who developed lun
of other ailments, and had to be put down in 2003
a sheep) of six. Some scientists suggested that t
wears out, meaning that cloning has less in comm
with the old ditto machine in the office of your
stopped enterprising entrepreneurs from cloning
tably, champion racehorses. And as of this writing
ings and Clone (don't look at us), will create a car
$32,000.

All these precedents have made it clear that the r
beings is now within the realm of the possible. W
another story, and a hot topic. Duplicating human
nerving, Frankenstein (or at the very least Boys fr
makes many people queasy. Though there have
cloned humans (despite a number of unsubstantiate
United Nations General Assembly adopted a large
lution to prohibit all forms of human cloning.

But cloning isn't only, or even chiefly, for reprod
tists are far more interested in the fact that clonin
cells, the jacks-of-all-trades of the cell world. Stem
pair system, dividing without limit and replenishin
These remarkable creatures can either stay as they

SCIENCE 541

adulthood, they were no longer
e organism).
or cloning—is pretty simple, at
al. The nucleus, along with the
mpty egg cell without a nucleus.
he egg and the nucleus are then
thed in a kind of chemical love
t has been fertilized, and in the
ough we like to think of clones
ed in a test tube, the fact is that
ppen in vitro. Soon after its cre­
t develops after conception) is
lly—or her cellular beginnings,
s before they were inserted into
d her to term.)
sly ineffective. For every happy
, according to some estimates,
ly, those aren't great numbers.

are genetically dysfunctional.
or have died prematurely. This
ng disease along with a number
at the relatively young age (for
the genetic blueprint actually
mon with a Xerox copier than

junior high. Still, that hasn't
other animals including, no­
g, one company, Genetic Sav­
rbon copy of your cat for only

reproductive cloning of human
hether or not it is desirable is
ns in the laboratory has an un­
rom Brazil) aspect to it that
been no successful births of
ed claims), in 2004 an alarmed
ely symbolic, nonbinding reso­

ductive purposes. Many scien­
ng can be used to create stem
m cells are part of the body's re­
ng other cells in the organism.
y are or morph into something

542 AN I N C O M P L E T E

more specialized, such as a mu
in this kind of technology—ca
ing diseases such as Alzheime

Unfortunately, therapeutic
lapsed in the public debate—p
conception. (To the latter, ex
destroys the embryo in the p
complexities have resulted in
where there is a mandatory
tempting to clone human bei
for the creation or manipula
have frustrated many scientist
Nancy Reagan, who became
her husband succumbed to Al
the key to future advancemen

One popular scenario warn
stem cells are inserted into an
in a living system. The resear
test the technology on lab rat
do we make of the chimera (n
part lion, part goat, and part
human qualities (at least on a
And what happens if a male la
male lab-created mouse that
there is the chance we could h
and not so cute."

E EDUCATION

uscle cell, a brain cell, or a red blood cell. Advances
alled therapeutic cloning—may hold the key to cur­
er's, Parkinson's, and diabetes.

cloning and reproductive cloning are often col­
particularly by those who believe that life begins at
xtracting stem cells from cloned embryos—which
process—is simply high-tech abortion.) The moral
n restrictions in many countries, including Japan,
ten-year sentence for anyone found guilty of at­
ings, and the United States, where federal funding
ation of embryos is prohibited. These restrictions
ts and their supporters (including former First Lady
an outspoken advocate of stem cell research when
lzheimer's), who see embryonic stem cell research as
nts in medicine. But even advocates urge caution.
ns of the experimental procedure in which human
n animal embryo to see how they divide and change
rch may be necessary—you would probably want to
ts before you tried it on a human friend—but what
named after the creature in Greek mythology that is

serpent) that has been created? Since it possesses
cellular level), does that mean it has human rights?
ab-created mouse with human cells mates with a fe­
also has human cells? As one science writer put it,
have "a sort of Stuart Little scenario, but in reverse

Making a Name for
in Science

EIGHT BRIGHT IDEAS TH

THEIR OWNERS'

ARCHIMEDES' PRI

Holds that buoyancy is the loss of weight an obj
placed in a liquid, and that that loss is equal to th
places. Takes off from (but is not restricted to) the

To recap, it's the third century B . C . ; we're in Sy
an important Greek outpost); and the local king, H
that the royal jeweler has sneaked some silver into
percent gold, royal crown. Hiero calls in Arch
stumped. He knows that gold weighs more than si
volume and that the volume of a piece of pure gol
weighing the same amount would be the same. Bu
of a strange-shaped thing like a crown? Then, pond
the tub, Archimedes realizes that a body immerse
liquid displaces exactly its own volume of that liq
Measure the volume of the water that's spilled over
side of the tub and you've got the volume of the t
in the tub. A t which point, Archimedes shouts "
reka!" and runs home naked, where he puts first
piece of pure gold, then the crown said to be of
gold, in a basin of water. The crown causes the wate
rise higher, revealing itself to have a greater vol
(and hence to be less dense, i.e., not pure gold). A n d
vealing the jeweler to be guilty.

The story told, bear in mind that Archimedes' p
ciple, as opposed to Archimedes' bath, applies no
water but to bodies floating on it, and not only to s
It explains, in addition to why ships float, why ball
determining what will and what won't sink, float,
volume must be considered, not to mention shape
age to remember anything here beyond "Eureka!
gravity," the ratio of a given density of a solid or a

SCIENCE 543

r Yourself
e

HAT LIT UP

LIVES

NCIPLE

ject seems to incur when it is
he weight of the liquid it dis­

"Eureka!" business.
yracuse, in Sicily (for centuries
Hiero II, has reason to suspect
o the new, and supposedly 100
himedes, who for a while is
ilver and consequently has less
ld and of a crown of pure gold
ut how to measure the volume
dering the problem one day in
ed in
quid.
r the
thing
"Eu­
t the

pure
er to
lume
d re­

prin­
ot only to bodies immersed in
solids but to liquids and gases.
loons rise, and it warns that in

or fly away, both weight and
and position. If you can man­
!" you might go for "spécifie
a liquid to the density of water

544 AN I N C O M P L E T E

(and of a gas to air), and a term
sums up what his principle w

FIBO

H e started with 1, as who am
he added the two: 1, 1,2. A
of what was now, unmistakab
And so on, all the way to 1
stop there? So . . . 233, 377,
you're wondering what, ap
thirteenth-century Pisa, acc
that he was working out the
bits can be produced from a
begets a new pair, which fro
suming that none of the rabb
a male and a female.

Wait, it gets interesting. Tu
with controlled population gr
instance, the ratio of scales di
5:8; of bumps around a pine
21:34. All of which are adjace
culture. Seems the ratio betwe
roughly 1:1.618—none other
tion, in mathematics the divi
whole segment is to the larg
Golden Rectangle, whose len
divided Golden Section-styl
proportions are so satisfying
tell it, both static unity and d
cade of the Parthenon, but Go
by everybody from Leonardo

THE LIN
TAXONOM

What's it to you whether or n
for cataloguing plants and ani
scientists who, up until the m
thing like "that little yellow f
wanted to compare notes. It

E EDUCATION

m that, while unknown to Archimedes, pretty much
inds up being all about.

ONACCI SERIES

mong us wouldn't? Then he repeated it: 1, 1. Then
nd he kept on adding, always the concluding pair
bly, a series: 1, 1, 2, 3; 1, 1, 2, 3, 5; 1, 1, 2, 3, 5, 8.
1, 1, 2, 3, 5, 8, 13, 2 1 , 34, 55, 89, 144. And why
610, 987, 1,597, but you get the picture. Now, if
part from the lack of late-night television in
ounts for Fibonacci's perseverance, we can report
solution to the problem of how many pairs of rab­
a single pair of rabbits, if every month every pair
om the second month on is itself productive, as­
bits die (or become bored) and each pair consists of

urns out that the Fibonacci series not only connects
rowth, it also keeps popping up on nature walks. For
istributed in opposing spirals around a pine cone is
eapple, 8:13; of seeds in the center of a sunflower,
ent Fibonacci pairs. If nature doesn't do it for you, try
een any two adjacent Fibonacci numbers (after 3) is
r than the ratio behind the celebrated Golden Sec­
ision of a line segment into two parts such that the
ger part as the larger part is to the smaller. The
ngth and width are the two parts of a line segment
e, is a big deal in art and architecture because its
to the eye, incorporating, to hear the aestheticians
dynamic variety. The classic example here is the fa­
olden Rectangles have also been found in paintings

to Mondrian.

NNAEAN SYSTEM OF
MIC CLASSIFICATION

not we have an orderly, scientifically sound method
imals? N o t much. But it comes in awfully handy for
middle of the eighteenth century, had to say some­
flower with the spots on its petals" every time they

was the Swedish botanist Carolus Linnaeus who

came up with the idea, shocking in its day, of divi
plants into twenty-four "classes" distinguished by t
sex (for which he has been called "the Freud of
botanical world"), that is, according to the length
number of stamens and pistils in their flowers,
subdividing those classes into "orders" based on
number of pistils. The sexual approach made sens
light of the then-novel idea of dividing God's creat
into broad categories, called "species," accordin
their individual characteristics; Linnaeus grouped
species by their ability to reproduce the same cha
teristics, generation after generation.

Besides coming up with a game plan, Linnaeus
mial nomenclature—that is, two names, one fo
one for the species (as in "sapiens")—as a shorthand
cumbersome and confusing descriptive names bot
Then he got everyone he knew to run around frant
mals as fast as they could, hoping to get absolutely
before some unsuspecting botanist or zoologist so
for a different species, thereby messing up the syst
names were arbitrary, spur-of-the-moment affairs
unsystematic and not all that helpful.

Modern naturalists also disagree with certain of
the old "kingdoms" of Plantae (plants) and Anima
third, Protista, to cover single-celled amoebas, b
like. Nevertheless, the Linnaean system, arbitrary
still the best we've got.

BROWNIAN MOVE

The zigzag, irregular dance done by minute partic
in a liquid; named for Robert Brown, the botanist
in 1827, while watching microscopic pollen grains
noticed that although the direction any particular
unpredictable, all the grains moved faster when th
down as it cooled. Einstein later did a paper on Br
that the grains were always in motion because they
water molecules; the hotter the water, the faster th
the more direct hits to the pollen grains. Eventua
came an important substantiating factor of the kin
ter, which states that matter is composed of tiny p
are constantly in motion.

iding
their
f the

and
then
n the
se in
tures
g to
d his
arac­

hit on the idea of using bino­
or the genus (as in "Homo"),
d labeling system to replace the
tanists were using at the time.
tically naming plants and ani­
y everything in creation named
omewhere used the same name
tem. As a result, many of these
s that may, today, strike us as

f Linnaeus' categories, and, to
alia (animals) they've added a
bacteria, slime molds, and the
y and artificial as it may be, is

EMENT

cles of matter when suspended
t who established its existence
s float around in water. Brown
r pollen grain would take was
he water got hotter and slowed
rownian movement, theorizing
y were being batted around by
e water molecules moved, and
ally, Brownian movement be­
netic molecular theory of mat­
particles (a.k.a. molecules) that

546 AN I N C O M P L E T

THE D

The change in the frequency
curs whenever there is a cha
ceiver; named for the early
Doppler. If the source of the
(or one is approaching the
wavelengths increases and
sounds and bluish light. If
sound waves are pitched low
cited example of the Dopple
tance, dropping in pitch as i
Used in radar to track the ve
distances between and rotat
track satellites. When, in 19
light from distant stars was
"red shift" to mean that the
sion, known as Hubble's La
verse is expanding.

BOO

Ever since Aristotle—as Geo
matician, would have been th
way of truth. That is, the way
tential for hysteria, tends to m

Boole's solution: Get rid
and by x and v, P a n d Q, what
as they're precise) stand for t
incidentally, often turn out a
nipulate the symbols mathem
such simple operations as ne
mean by "not"), conjunction
where you're supposed to stan
ter how complicated or abstr
"0" (standing for "nothing" o
but also the device known a
tions of true and false value
statements) and the intellect
ceeded in wresting logic fro
mathematicians).

For fifty years nobody but

E EDUCATION

DOPPLER EFFECT

y of a wave (whether of sound or of light) that oc­
ange in the distance between the source and the re­
y nineteenth-century Austrian physicist Christian
e waves and the receiver are approaching each other

other), Doppler observed, the frequency of the
the waves get shorter, producing high-pitched
the source and receiver are moving farther apart,
wer and light appears reddish. (The most commonly
er effect: the train whistle that screeches in the dis­
it approaches the platform where you're standing.)
elocity of a moving object; in astronomy to measure
tions of stars, planets, and entire galaxies; also to
929, the astronomer Edwin Hubble noticed that the
becoming redder, he took this "Doppler shift" or
e stars were rushing away from earth. His conclu­
aw and now generally accepted, was that the uni­

OLEAN ALGEBRA

orge Boole, the nineteenth-century English mathe­
he first to tell you—language has been getting in the
y we talk, with all its inaccuracy, ambiguity, and po­
mess up the way we think.
o f the words altogether, instead letting symbols (a
tever; it doesn't matter that they're arbitrary as long
the components of thought, which, not entirely co-
also to be the elements of formal logic. Then, ma­
matically, in a kind of mental algebra that's based on
egation (corresponding roughly to what you and I
(our "and"), and alternation ("or"); and that—here's
nd up and applaud—always reduces things, no mat­
ruse, to either a " 1 " (standing for "all" or "true") or a
or "false"). Thus was not only Boolean algebra born,
as the truth table (which lists all possible combina­
es that can accrue in the interplay of two or more
tual specialty known as symbolic logic (which suc­
om the philosophers and delivering it over to the

Lewis Carroll seems to have gotten all that turned

on by Boole's "laws of thought," but those laws d
I, knock the socks off Alfred North Whitehead
Russell, who relied on them to establish more or l
was logic the proper domain of mathematics, it
roots lay. Since then there's been no stemming th
took to constructing truth tables in the courtroo
engineers began thinking of parallel "on/off" sw
cuitry in O-vs.-l terms. To the point that today
doesn't proceed along the relentlessly binary lines
stunning an example as the atom bomb of how the
ter, fuels technology.

MÔBIUS STR

Take an ordinary flat strip of paper. Give it a half
two ends together to form a loop. O K , ready? Take
in one side of the loop, and a green Magic Ma
Whoops! That's right: The strip has only one "side
Easy to construct and hard to imagine, the so call
the nineteenth-century German mathematician a
scribed it) also reacts strangely to scissors: Cut it a
down its middle and you'll get not two Môbius s
strip twice as long as the one you started with. (The
for this: A Môbius strip has only one edge; the
with it a second side.) Now try cutting a new strip

SCIENCE 547

did, shortly before World War
(see page 328) and Bertrand
less persuasively that not only
was indeed where the latter's

he Boolean tide. First lawyers
m. Then telephone-company
witches and "open/closed" cir­
y its the rare computer that
s first laid down by Boole—as
eory, and the wackier the bet­

IP

f-twist. Now Scotch-tape the
a red Magic Marker and color
arker and color in the other.
e"—or do we mean "one" side?
led Môbius strip (named after
and astronomer who first de­
along a line drawn lengthwise
strips but a normal two-sided
e mathematicians' explanation
cut adds a second edge and
p along a line one-third of the

548 AN I N C O M P L E T E

way in from its edge; we won
weird, too.

A s to what's really going on
dertaken in space, collectively
cises in paradox reminiscent
nevertheless crystallizes a bun
case of the Môbius strip, wha
given point to be two-sided,
sided? One theory: that life i
ing to consider everything i
collection of neurons, say) ris

GÔDEL'S INCO

The problem here—if you,
want to look at it as a problem
logicians. In 1931, the Czec
that within any given branch o
sitions that couldn't be prove
(statements like 1 = 1 that a
branch itself. You might be
numbers within a system by
new rules and axioms, but by
own unprovable statements.
complexity are, by definition,
time, more true statements th
ing set of rules.

Godel's theorem has been u
as a human being because the
axioms, whereas people can d
pocket calculators that are sm
school with.) It plays a part in
power of language to come u
been taken to imply that you
mind, like any other closed sy
by relying on what it knows.

E EDUCATION

nt tell you what you'll wind up with, but it's pretty

n here: The subject is geometric transformations un­
y known as topology. Superficially a series of exer­
of all those old M . C . Escher drawings, topology
nch of the century's top issues. For instance, in the
at does it purport that something that appears at a
when traced round its continuum, is in fact one­
is holistic, not reductionistic; that persisting in try­
in terms of its component parts (the brain as a
ks missing the whole (the brain as mind).

OMPLETENESS THEOREM

unlike most twentieth-century theoreticians, still
m—is self-reference, a historic stumbling block for
ch-born mathematician Kurt Gôdel demonstrated
of mathematics, there would always be some propo­
en either true or false using the rules and axioms
are accepted without proof) of that mathematical

able to prove every conceivable statement about
going outside the system in order to come up with
doing so, you'll only create a larger system with its
T h e implication is that all logical systems of any
, incomplete; each of them contains, at any given
han it can possibly prove according to its own defin­

used to argue that a computer can never be as smart
e extent of its knowledge is limited by a fixed set of
discover unexpected truths. (Maybe, but we've met
marter than some of the people we went to high
n modern linguistic theories, which emphasize the
up with new ways to express new ideas. And it has
u'll never entirely understand yourself, since your
ystem, can only be sure of what it knows about itself

Fun—or at Least
Minutes—with N

Either mathematics is poetry, all density and pr
grant type of entrepreneurship, in which nobo
quarters is taller than anybody else's headquarters
symbols we use to count up all the things around us
uous—an area where mathematicians have gotten ei
pacious over the years, depending on how you loo
number "types" they've managed to corner and/or c

The Number System
*Any of these rings—depending
on whom you listen t o —
define the limit of
K 0 . i-e.,
normal
infinity,

Each circle
contains all
the ones within it.
The algebraic numbers
include all but the
transcendental ones.
"The two applicable transfinites.

SCIENCE 549

a Few
Numbers

recision, or it s the most fla­
ody's satisfied until his head­
s. Exhibit A: Numbers—the
s that aren't unique or contin­
ither more lyrical or more ra­
ok at it. Below, some of the
construct for themselves.

*This ring defines the limit of Ki—
the real numbers,
which together
constitute the
Continuum,
C.

Normal"
(= Algebraic)N
Irrationals

V3

^7,^80
.2307024501

the
Complex
Numbers

2 + 3/ ,N
V 2 + V3/
the
Quaternions
4 + 2 / - 3 / + 5/C
2 + V3i+%j+Trh

55° AN I N C O M P L E T

NATURAL

NATURE: The counting numbers.

EXAMPLES: 1, 2, 3, 4 , . . .

HISTORY: In use since man first felt a need
sues o f Penthouse—he owned. S
"two," and "many."

PRACTICAL USES: Conducting censuses, scoring ba

MATHEMATICAL One of the first creative highs.
RESONANCES.* were on to something big, somet
tinctions between "odd" and "ev
and "ordinal" (1st, 2nd, 3rd; the
finity," the quality of endlessness
bers were seen to progress.

PRIME

NATURE: Those natural numbers higher t
other than themselves and 1. (O

EXAMPLES: 2, 3, 5, 7 , 1 1 , 1 3 , 1 7 , . . .
HISTORY:
Arduous. Also, odd. Euclid prov
constructed a "sieve" to isolate t
tried to devise a formula that wo
the Goldbach conjecture of 1,74
7,816,230 digits; needless to say

PRACTICAL USES: None whatsoever, unless you fin

MATHEMATICAL A thorn in the side. Mathemat
RESONANCES: primes fall where they do, or,
primes are just one example of
Greeks and others managed to
fects and the amicables.

E EDUCATION

d to know how many of something—goats, pots, is­
Some primitive societies still make do with "one,"

asketball games, seating people at dinner parties.
The sense among early "numbers" types that they
thing real and abstract. Also, the inevitability of dis­
ven," and "cardinal" (1, 2, 3; the counting numbers)
ranking numbers). Ditto, of considerations of "in­
s, symbolized as °°, toward which the natural num­

than 1 that can't be evenly divided by any number
Opposed to the composite numbers.)

ved there was no end to the primes. Somebody else
them. Any number of Renaissance-and-later men
uld generate all possible ones: no dice. Then there's
42. The largest known prime (as of this writing) has
y, they now have the computers on the case.
nd yourself at a mathematicians' convention.
ticians can't bear that they still don't know why
for that matter, how many there are. Then, too,
sets of numbers having special properties that the
isolate. Another time we'll tell you about the per­

INTEGER

NATURE: The natural numbers, plus 0 an
numbers.

EXAMPLES: . . . , - 4 , - 3 , - 2 , - 1 , 0, 1, 2, 3, 4

HISTORY: Zero, an ancient Hindu innovat
the Arabic numerals; it was an
gare, "to deny") had a harder ti
Greeks, so flexible in other resp
the sixteenth century did an Ital
pointing out that there can be le

P R A C T I C A L U S E S : Negative numbers make debit-h
from 307 or 3,700.

MATHEMATICAL Now there was a genuine numb
RESONANCES: Also, all equations of the A + x
all its inherent weirdness, to thi

RATIONAL

NATURE: The integers, plus all fractions (
ative, a fraction being defined a

EXAMPLES: V2,7A, - 2 A , .25, .3333,...
HISTORY:
The Egyptians and the Babylon
know what they were doing, an
a number like 7V2, which was n
nify, codify, and promote the id

PRACTICAL USES: Measuring (as opposed to coun
Brie, and drapery material, whi
sharing.

MATHEMATICAL A flush of pleasure: Rational
RESONANCES: equations, of the Ax + B = 0 var
metrically speaking, all you need

REAL The rational numbers, plus the
in fraction form: imperfect squ
NATURE: terminate nor repeat; the transc

SCIENCE 551

nd all the negative numbers. Also called the whole

4,. . .

tion, arrived in Europe in Roman times, along with
instant hit. Negative numbers (from the Latin ne-
ime: If the root of an equation was less than 0, the
pects, called it "fictitious" and threw it out. Only in
lian, Cardano, systematically use negative numbers,
ess than nothing—a debt, for instance.

heavy bookkeeping elegant. Zero makes 37 different

ber line, stretching infinitely to both left and right.
= 0 variety could be solved. And, there was 0, with
ink about.

(or their decimal representations), positive and neg­
as the ratio of two integers, a/b, where b 4 0.

nians could handle fractions, though (a) they didn't
d (b) conservatives of the day sneered at the idea of
neither 7 nor 8. A s usual, it took the Greeks to dig­
dea. Since then, second nature to everybody.
nting), dealing with continuous quantities like age,
ch don't always break into convenient pieces. Also,

ls felt g o o d . B e s i d e s , all linear (or first-degree)
riety, now had one—and only one—solution. G e o ­
ded was a straightedge.

zillions of irrational ones, which can't be expressed
uare, cube, and higher roots; decimals that neither
cendentals.

551 AN I N C O M P L E T

E X A M P L E S : ft,3 ft , 19y3ÏÏ ; .23020300200

HISTORY: ft, the first irrational, was dis
structed a right angle, each of w
potenuse. His proof that it coul
and resulted in the immediate sa
cumference to its diameter, wasn
was discovered that there are a w

PRACTICAL USES: W i t h the exception of TT and e,
hand, square roots are as much a
the march from Carmen.

MATHEMATICAL Like arriving at your surprise pa
RESONANCES: But, the real numbers could be
with all the points on a line (rega
termed the Continuum (abbrevi
the mere °o formed by the natura
(or second-degree) equation of t

TRANSCENDENTAL

NATURE: A special category of irrational
gebraic.

EXAMPLES: TT, e, plus all the trigonometric
quantity (except 0 and 1) raised

HISTORY: In 1844 the Frenchman Joseph
irrationals that would not serve a
mial equations in any of the infi
up with an example. In 1873 so
"natural" logarithms, was transce
same of TT. Since then, so many
simply stopped answering the do

PRACTICAL USES: TT (or pi, from the first letter of p
dispensable in carpentry and con
and, in the days before slide rule
ness, enjoyed a certain vogue. Th
gauge the height of a telephone
live all the phone lines are unde

E EDUCATION

000732 . . .; TT, e

scovered by Pythagoras, c. 500 B . C . , when he con-
whose sides was one unit long, and measured its hy-
ldn't be put into fractional form was very upsetting,
acrifice of a hundred oxen, TT, the ratio of a circle's cir-
n't proved to be irrational until 1761. In between, it
whole lot more irrational numbers than rational ones.

, few if any, despite all the brouhaha. O n the other
a part of the collective consciousness as asparagus or

rty with a 102° fever: elation tempered by shakiness.
shown to correspond in a perfect, one-to-one way
ardless of its length); this endless series of points was
iated C), and proved to be much more endless than
al numbers. Also, you could now solve any quadratic
he Ax2 + Bx+ C = 0 variety.

numbers: They're real, all right, but they're not al-

c ratios and logarithms they give rise to; also any
d "radically," e.g., 2 v I .

Liouville proved the existence of transcendentals—
as solutions to any of the infinite number of polyno-
nite number of degrees possible—but couldn't come
mebody else showed that e, the base of the so called
endental. In 1882 still another somebody showed the
transcendentals have turned up that a lot of us have
oorbell.

perimetron, Greek for "measurement around") is in-
nstruction, e figures in statistics and nuclear physics,
es and pocket calculators put logarithms out of busi-
hey tell us that trigonometric functions allow you to
e pole from the length of its shadow, but where we
erground.

MATHEMATICAL The transcendental numbers co
RESONANCES: ones. Also, since TT is transcend
only a straightedge and compas

COMPLEX

NATURE: T h e real numbers, plus all the im
cially, these numbers when seen

EXAMPLES: 3 + 2i, 3 - Jïi, 0 + Jïi, 3 + 0/

HISTORY: What do you do with an equatio
prolific mathematician of all ti
stand for the J-I. Like the X in
mission of perplexity. But imagi

P R A C T I C A L U S E S : The description and handling o
but direction.

MATHEMATICAL The complex numbers constit
RESONANCES: one way with the points in it
board, you'll always have the so
Cx"~2 + . . . + Z = 0 variety, no ma
the same number of solutions
solve any such polynomial, prov
ematicians were so excited by al
result was quaternions (see next

ALGEBRAIC

NATURE: So called because they turn up
means all of the above numbers

EXAMPLES: To reprise: 2, 0, - 2 , V2, 3A, .33
-3i

HISTORY: The nineteenth century was th
Ideals, unique factorization, and

P R A C T I C A L U S E S : There were repercussions in mat
the living-room lamp. Frankly, l

MATHEMATICAL First and foremost, a way of g
RESONANCES: make sense and can breathe a b
vestigating the solution of prob
self. T h e beat goes on.

SCIENCE 551

onstitute an infinity even greater than the algebraic
ental, nobody has to try to "square the circle" using
ss anymore. Flash: el = - 1 .

maginary ones, based on i, defined as ftï. But espe-
to consist of both real and imaginary components.

on like x2 + 1 = 0? In 1777, Leonhard Euler, the most
ime, introduced the symbol i (for "imaginary") to
n "X-ray," it was both a shriek of victory and an ad-
inary numbers, like negative numbers, caught on.

of vector quantities, which have not only magnitude

tute a plane and correspond in a perfect one-to-
(cf. the line of real numbers). Also, with them on
olutions to polynomial equations of the Ax" + Bx"'1 +
atter how high n gets, and you'll always have exactly
as the degree o f n. W i t h the complexes, man can
viding it has rational coefficients. In general, math-
ll of this that they couldn't not push their luck: T h e
t page).

as the solutions of algebra-style equations. Which
except the transcendental ones.

333 . . . , .2302030020000732 . . . , ft y3 + 2i, 0.5

he big one here: Gauss, Kummer, and Dedekind.
d Abelian number fields.

thematical physics, but we admit that's not rewiring
lingering over this category is probably a mistake.

gathering all of the above together so that they
bit. However, what was originally a scheme for in-
lems had now become a whole thing, an end in it-

554: AN I N C O M P L E T E

TRANSFINITE

NATURE: "Styles" of infinity. And the sky

EXAMPLES: K0, Ki, K2, K3, . . . , Kn (The
"aleph-null," "aleph-one," "aleph

HISTORY: In 1895, the German mathemat
infinity," a whole series of endle
o°, roughly equivalent to the en
tems, the integers or even the r
uum, C , roughly all the real num
points in a line (or a square or a
ness of all curves. And nobody
alone aleph-thirty, might stand

PRACTICAL USES: Hmmm?

MATHEMATICAL The mathematical imagination
RESONANCES: out for Indian food. While every
Cantor's baby, and he's with his

QUATERNION

NATURE: The complex number concept e
EXAMPLES:
4 - 2i - 3 / + 5k, 2 + j57+ %j- i

HISTORY: First suggested by William Ham
with complex numbers. Subseq
you suspected they would be.

P R A C T I C A L U S E S : Engineers get into them. But yo

MATHEMATICAL The feeling that, once having l
RESONANCES: here is that when you extend nu
expense of something called p
granted fall away. For instance,
role 0 plays or multiplicative com
x). Say good night, Gracie.

E EDUCATION

y's the limit.

e letter is aleph, first of the Hebrew alphabet. Say
h-two," and so on. Or maintain a discreet silence.)

ician Georg Cantor worked out "the arithmetics of
ssnesses. Aleph-null is simple infinity: the familiar
dlessness of the natural numbers (or, in some sys­
ationals). Aleph-one is the infinity of the Contin­
mbers or, depending on how you look at it, all the
a cube). Aleph-two may or may not be the endless­
y has been able to figure out what aleph-three, let
for.

n rents the video o f Yellow Submarine, then goes
ybody continues to talk about this one, it was really
alephs now; a kind of dead end.

xtended from two dimensions to four.

irk

milton in 1843 in response to his personal successes
quently taken into even more dimensions—just as

ou're up past your bedtime.

learned to walk, one can run. Also fly. T h e lesson
umbers beyond the complex stage, you do so at the
ermanence; one by one, properties you took for
, with quaternions, you have to give up either the
mmutativity (i.e., x times y no longer equals y times

Double Wham

A s it happens, two of the biggest deals in mod
ter e\ entropy and evolution. Each spans two
each has the kind of reverberations that can't reall
room walls. These are their stories.

ENTROPY, THE LA
THERMODYNAMICS, AND

HAVE BEEN FEELING T
LISTLESS LATE

Entropy is what the Second Law of Thermodynami
most of us seem to have gotten bogged down short
ing surprising about that—understanding the First
the way of concentration or stamina. The basic prin
branch of physics dealing with the transformation
forms of energy, it simply states that "energy is cons
tible—there is always the same total amount of it in
of Thermodynamics was a big deal back when peop
steam engines; today, it serves chiefly as the solid p
and philosophers like to hurl themselves into the a

The latter states, with deceptive simplicity, that
tends to a maximum." Already, whether you know it
tropy is a measure of the total disorder, randomne
thermodynamics, it crops up every time any work
work ever gets done is through heat transfers—ho
with cool air, for instance, to produce the steam th
the outset, the system is said to be at a low level
water molecules are distinct from the slow-moving
thing has a kind of order to it. But as the heat flows
heat naturally does—the fast-moving molecules beg
ing with the slow ones until eventually all the mole
mately the same speed. At this point, we're at maxim
the same temperature, all the molecules are milling
order, and nothing more can be accomplished—the
uratively, run out of steam. T h e energy within the s
you separate the molecules again, returning them to
some and chilling others, it can't be used to make t

SCIENCE 555

mmy

dern science start with the let­
centuries (plus this one), and
ly be done justice to on rest-

AWS OF
WHY YOU MAY
TIRED AND
ELY

ics is all about. Unfortunately,
tly after the First Law. Noth­
Law doesn't demand much in
nciple of thermodynamics, the

of heat into work and other
served"; that is, it's indestruc­
n the universe. The First Law
ple were trying to build better
platform from which scientists
abyss of the Second Law.
t "the entropy of the universe
t or not, you're in trouble. E n ­
ess, or chaos in a system. In
gets done, since the only way
ot water coming into contact
hat drives a steam engine. At
of entropy: The fast-moving
g air molecules, and the whole
s into the cooler medium—as
gin to spend themselves, mix­
ecules are moving at approxi­
mum entropy; everything is at
g about without any particular
e system has, literally and fig­
system is still there, but unless
o a state of tension by heating
hings happen anymore.

556 AN I N C O M P L E T E

Not your problem, you thin
for-brains. Outside thermody
state of relative order to one o
what happens when you let y
before you know it the ice cu
worth dealing with is the lem
nice neighborhood and within
happens to your body from ab
in the nineteenth century, it's
to our galaxy—in the scenario
energy, the product of a hot
molecular chaos will prevail,
bath watching our toes decay.

You can see why the concep
with keeping their corset stay
brood about our weight and t
which has all sorts of interest
"arrow of time," a metaphor
tween past and present; the fa
from order to disorder, and no
two still photos of eggs, one
scrambled in a frying pan, yo
order and you know somethin
other thing about progressiv
plexity. The picture of the eg
make sense of than the one s
cepts—the arrow of time and
helped convince scientists ear
have a beginning, and encou
Bang (see page 509).

Entropy is also a hot topic
increased complexity seems to
leads to richness of thought a
cording to the Second Law,
not. But maybe that's becaus
apparent order we see in our
other extremity? Then there's
tropy is embroiled in a way t
There is no law that states a
scrambled eggs can't re-form
adolescent glory—only that th
as to be virtually nil. At leas

E EDUCATION

nk, since your car runs on gas? Guess again, mush-
ynamics, increased entropy—things going from a
of disorder—is the upshot of all natural actions. It's
your vodka on the rocks sit around for a while and
bes have melted into the vodka and the only thing
mon twist. It's what happens when you move into a
n a few years it turns into the South Bronx. It's what
bout age thirty on. And, as scientists predicted back
what's going to happen to the universe—or at least,
o known as the "heat death" of the universe: Solar
sun turning in cold space, will inevitably run out,
and we'll all be left sitting in a lukewarm cosmic
.
pt of entropy made the Victorians, already obsessed
ys in place, rather sad. Today, however, we prefer to
take a more objective approach to the Second Law,
ing ramifications. It has, for instance, given us the

that expresses the purely physical distinction be­
act that time is an observable, one-way progression
ot just a figment of our imaginations. If you look at
showing them in their unbroken shells, the other,
ou immediately know which came first; reverse the
ng's screwy. T i m e flows in one direction only. An­
e disorder: It's synonymous with increasing com­
ggs in their shells is neater, simpler, and easier to
showing a gloppy mess in the pan. These two con­

the increased complexity of high-entropy states—
rlier in this century that the universe did, in fact,
uraged them to come up with the idea of the Big

c among information theorists, who point out that
o add up to more, not less, order; and anyway, if it
and communication, more power to it. Besides, ac­
the world should be falling apart by now, and it's
se the universe is expanding; who's to say that the

corner of it isn't balanced by total chaos at some
s the whole business of probability, with which en­
that's enough, in itself, to make your brains hurt:
absolutely that entropy cant be reversed, that the

into their shells or your body revert to a state of
he probability of any of that happening is so small
st in the universe as we know it. But what do we