Gardner M. Sixth Book of Mathematical Diversions from Scientific American

Mathematical Games

correspondence with every point inside.

A+I

line extending outward from the board

to infinity corresponds to

a

line inside the

central

white space, extending inwitrcl

toward the center but never reaching it.

Inversion geometry can, of course, be

applied to three-space as easily as to

the

plane. An old nlathernatical joke says that

to catch

a

lion you just t)uild a cage and

perform an in\~ersion operation

011

the beast.

The cosmos itself can be inverted

and corn-

pressed inside

a

tennis ball. In this country

during the 1870's

a

religious cult was ac-

tually founded on the belief that such an

inverted three-space reflects the true state

of affairs.

Cyrus Reed Teed's "Koreshanity"

put the entire

uni\,erse iil.~idc the earth.

\Ye imagine ourselves on the outside of

the earth looking out at gigantic stars

scattered through an

infinite space; the

truth, said Teed, is that we are on the in-

side of

a

hollo\v earth looking

ill

at srnall

stellar bodies moving in a space that is the

geoinetrical inverse of the space of ortho-

dox astronomy. Teed defended his views

in many books and articles; years later his

ideas attracted

a

following in Nazi Ger-

many. (For

inore details on this crazy cult

see Chapter

2of my

Fuels

c~nd

Fallacies

iiz

tlie S(~il~e

of

Scieilce; New York: Dover,

1957.)

Figure 176 shows two exa~llples of rrlany

vertigo-inducing patterns that were studied

by psychologists

more than

50

years ago.

They are kilowrl as "twisted-cord illusions"

because they were first discovered by

twist-

ing l~lack ;ind white string into

a

single cord

that was then arranged in various ways

011

differently patterned backgrounds. The top

part of the figure consists of concentric

circles (as you can prove

with a compass);

in the one at the

bottorn a spiral is made up

of straight horizontal

and vertical "cords"

(as you can prove with a ruler).

Tesselations of the

plane created by

fitting together replicas of the same basic

shape have long been used

in design and

are now turning up in many of the latest

013 fabrics. The cross-pentomino appears

on an

013 dress advertised in 1965 by Bon-

wit Teller.

All

polyominoes and polyia-

inonds (polyiamonds are formed by joining

equilateral triangles

instead of squares)

of order six or less will fit together to cover

the plane, but so far I have seen only the

cross-pentomino and the L-tromino (the

latter on a scarf sold by

Gimbels in New

York City) on

013 fabrics. The reader can

easily create his

own new Op patterns by

fillding ways to tile the plane with each of

the 12

pentominoes and the 12 hexiamonds

(for the hexiamond shapes see this book's

chapter on polyiamonds).

\lost, but not all, of the 108 heptominoes

(for their diagrams see Solon1o11

W.

C;olomb's

book

Polyonainoes,

pages 108-109) will tile

the plane. Several British mathematicians

are working on the difficult question of

which of

thern are

11ot

plane-fillers. The

corresponding problem for the

24 lleptia-

176.

"Twisted cord" concentric circles

(top) and spiral (bottom)

Mathematical Games

monds

[see

Figure

1771 was proposed by

T.

H.

O'Beirne and was solved this year

by Gregory

J.

Bishop of Boston. Only one

of the

24

shapes is not a plane-filler. Can the

reader identify it

and prove that it ca~lnot

tesselate the plane?

The Op pattern that covers the plane with

convex noncongruent heptagons

[Figure

1781 embodies a curious paradox that twid-

dles the brain even more than the eye.

If

this pattern is repeated infinitely, what is

the average angle in it? Since the plane

contains nothing but heptagons, and since

the interior angles of any heptagon

sum

to

900

degrees, it follows that the average

angle is

90017

or

1284/7

degrees. Note,

however, that every point on the pattern

is a meeting of three angles. This surely

requires that the average angle be

36013,

or

120

degrees. Explain!

177.

The

24

heptiamonds.

Which cannot tile a plane?

pva

ITW

P

WAU

+

++*

178.

Tesselation

of

convex heptagons

Answers

corltinuously distorted to produce a check-

erboard is evident from the fact that the

Were you able to make one straight cut

number of its cells,

392,

is not a square.

across the circular

Op

pattern

[see

tlie

Note also that the two cells inside the

drazoirrg

at left

in

Figure

1791

so as to bull's-eye are each three-sided; any dis-

divide the pattern into two parts, each tortion that turns one of these cells into a

topologically equivalent to a square check-

\quare would turn the other into

a

non-

erboard? That the pattern itself cannot be convex figure. It is therefore necessary

180.

Proof that the V-heptiamond will not tile the plane

that the cut separate these two cells. The

only straight cut that does this is one along

the horizontal diameter of tlie large circle.

The figure in the middle is

topologic;illy

the same as the one at the left. It is easy

to see that

a

single cut along

AB

produces

two halves, each of

which is topologically

the

same as

a

square checkerboard 14

cells on a side

[Figure

179,

right].

The only heptiamond that will not tile

the plane is

the \'-shaped figure shown at

a

in Figure 180. The proof, hy Gregory

J.

Bishop, an electrical engineering student

at Northeastern University, is simple.

A

second piece can

fill

the colored triangular

concavity of the first one

only as shown at

b.

(b7e ignore

a

mirror reversal of the second

piece.) The colored triangle of

the second

piece can now be filled only by placing

a

third piece as shown at

c.

The colored tri-

angle of this figure

illust in turn be filled

by placing a fourth piece as shown at

d,

and there is a similar lack of choice in

positioning pieces

5

and

6.

Now we are

stuck. There is no way to fill

the colored

triangle associated with piece

6.

Tessellations for tlie other

23

heptia-

monds are too numerous to illustrate. The

reader may have discovered the useful

trick of pairing two pieces to form a pattern

that periodically tiles the plane. For ex-

ample, five different

heptiarnonds can be

paired to fit the

sanle periodic tessellation.

(Figure

181

shows four; can the reader

discover the fifth?) Bishop has also estab-

lished that each of the

66

octiamoiids will

tile the plane, and that all but four of the

108

heptominoes will do so.

181.

Tesselation for four heptiamond pairs

The paradox of the heptagon tessella-

tion was taken from

Hugo Steinhaus'

One

Hundred Problems

in

Elemel~tary Alnthe-

matics.

The paradox arises from the fact

that a

rearrangement of terms in an infinite

series can lead to

a

different calculation

of the average term.

Steinhaus gives as

an example the series

1,

0, 1,0, 1,O

.

for

Mathematical Games

which

112

is the average. But the two in-

finite sets of ones and zeroes can also be

arranged

1,

0,

1,

0, 0, 0, 0,

1,

0, 0, 0, 0, 0,0,

0,

1

.

(where successive sets of

zeros have cardinal numbers that are squares

of

1,

2,

3 .

.

.),

in which case the average

is

0.

It is easy to form other arrangements

to make the average any desired integral

value between

0

and

1.

In the heptagon

pattern two different arrangements of two

infinite sets of angles are considered, and

there is no reason why the calculation of

an average angle

should be the same in each.

References

"Pursuit Curves and Mathematical Art."

I.

J.

Good.

The Muthemutical Gazette,

Vol. 43,

No. 343; February, 1959. Pages 34-35.

"Two Hexagonal Designs." Prakash Chandra

Sharma.

The Mathematical Guzette,

Vol. 45,

No. 351; February, 1961. Pages 26-27.

"Op Art: Pictures That Attack the Eye."

Time,

Vol. 84, No. 17; October 23, 1964. Pages 78-84.

"Op Art."

Life,

Vol. 57, No. 24; December 11,

1964. Pages 132-140.

"Art That Pulses, Quivers and Fascinates."

John Canaday.

The New York Times Magazine,

Vol. 114, No. 39,110; February 21,1965. Pages

12-59.

"Optical Art." Gerald Oster.

Applied Optics,

Vol. 4, No. 11; November, 1965. Pages 1359-

1369.

Optical

illusions

and the Visual Arts.

Ronald

G.

Carraher and Jacqueline

B.

Thurston. New

York: Reinhold, 1966.

Cc/I.)ernetic Serendipity: The Coi~zputer and the

Arts.

Edited by Jasia Reichardt. London:

Studio International,

1968. (Special issue.)

25.

Extraterrestrial Communication

Across the gulf of space, minds that are to our minds as ours are to those of

the beasts that perish, intellects vast and cool and unsympathetic, regarded

this earth with envious eyes, and slowly and sureiy drew their plans against us.

H.

G.

Wells, The War of the Worlds

IN 1898,

WHEN

WELLS'S

NOVEL

was first

published, a number of distinguished as-

tronomers seriously believed Mars was in-

habited by creatures with "intellects vast

and cool" and superior to our own. The

Italian astronomer Giovanni Schiaparelli

(the uncle of the dress designer Elsa

Schiaparelli) had reported in 1877 that he

saw fine lines crisscrossing the red planet. A

wealthy Bostonian, Percival Lowell, became

so excited by Schiaparelli's continued dis-

closures that he decided to abandon

Oriental studies and become an astronomer.

In 1894, when Mars was unusually close to

the earth, Lowell established his own

observatory on "Mars Hill" in Flagstaff,

Arizona.

Lowell too saw the lines Schiaparelli

had called "canali." (The word, which

means "channels," had been subtly

mis-

translated "canals.") Indeed, he saw them

in fantastic profusion; eventually he mapped

more than 500. In three books

-Mars

(1895),

Mars and Its Canals

(1906) and

Mars as

the Abode of Life

(1908)-Lowell argued

that the lines he saw were wide bands of

vegetation bordering enormous irrigation

ditches constructed to bring water from

melting polar caps to the dry Martian

deserts.

"That Mars is inhabited by be-

ings of some sort or other," he wrote, "we

may consider as certain as

it

is uncertain

what those forms may be." Lowell's Mars

books had an enormous influence on early

science fiction; the canals turned up every-

where, from Wells's 1897 short story "The

Mathematical

Games

Crystal Egg" to the later llartian romances

of Edgar Rice Burroughs.

There is no doubt about Lowell's compe-

tence as an astronomer. Calculations he

made in 1915 led to the discovery of Pluto

by Clyde

\V.

To~ilbaugh in 1930-at the

Lowell Observatory. ("Pluto" was chosen

as the planet's

name because its first two

letters are Lowell's

initials and its last

two the beginning letters of Tombaugh;

Pluto's

syrnbol,

E,

conlbines

P

and

L.)

But

as Jonathan Norton Leonard observes in

his book

Flight

illto

Spc~ce,

Lowell's tem-

perarnent was closer to that of his sister

Amy, the cigar-smoking poet,

than to that

of his cautious, conservative brother

Abbott Lawrence,

who became president

of Harvard. Although

some astronomers

e~lthusiastically confirmed Lowell's ob-

servations of

llartian canals, others with

better telescopes and better eyes could

see no canals at all. Even in today's best

telescopes hlars is a tiny, jiggling spot of

light, and in those rare, fleeting

niomellts

when the image holds still, one's mind

can play strange tricks. Photographs are

no help because the earth's turbulent

atmosphere blurs the

image.

The consensus aillong astronomers today

is that Scliiaparelli, Lowell and their fol-

lowers

were the victims of optical illusions

induced

by irregular splotches on the red

planet and elaborated by astigmatism and

psychological self-deception.

Among the few living scientists

who

continue to take Lowell's speculatioils

seriously the most vocal is

\Veils

Alan Webb,

a California

cheinist. In his book

31ars,

the

New

Frontier: Lowell's Hypot1zesi.s

(1956)

and

in many magazine articles and lectures

he

has reported on an interesting topologi-

cal

aiialysis of canal drawings made by

Lowell and by one of his leading sup-

porters, Robert

J.

Trumpler. Considering

the

maps of these two astrononlers as geo-

inetrical networks, Webb determined the

percentage of vertices at

which three, four,

five, six, seven, and eight rays came to-

gether. On the maps drawn by both men

vertices of order

4

(four lines meeting at a

point) predominate: they constitute about

43

percent of the vertices on Trumpler's

maps and about 35 percent on Lowell's.

A

similar analysis of networks found in

nature- mud cracks, shrinkage cracks of

glazed chinaware, cracks

in ancient lava

beds, rivers,

and so on-sho\vecl order-3

vertices leading their percentage list. Only

in networks constructed

by living things,

such as spider webs and animal trails, did

Webb find the order-4 points predominat-

ing. The networks that are topologically

most like the Lowell-Trumpler

maps are

such man-made ones as railroad lines and

air travel routes. Thus does topology,

Webb argues, back

11p Lowell's intuitive

conviction that the canals must have been

the work of high-order intellects.

Webb's arguments assume, of course, a

correspondence between the Martian sur-

face and the Lowell-Trumpler maps. But

if these maps are

no more than doodles of

what

Lowell and Trumpler imagined they

saw, their topological similarity to rail-

road

lines is easily understood. At this

writing the first television pictures of

hlars

Extraterrestrial Communication

are being received from

Afuri~zer

IV,

but

they have not yet established whether or

not there are lines on the planet. Certainly

few astronomers expect the pictures to

show anything like Lowell's cobwebs; if

they do, the great canal controversy

will

surely break out again.

Fro111 1880 to 1925, when interest in

hlartian canals was high, all sorts of pro-

posals

were put forward for establishing

contact with

llartians. Two frequent sug-

gestions were that a

potverful searchlight

be built that would blink a code message,

or that a chain of bright lights be stretched

across

a

vast area to make a diagram, visible

in Martian telescopes, of the Pythagorean

theorem. There was much discussion about

radio contacts: sending a series of beeps to

represent the counting numbers (beep;

beep, beep; beep, beep, beep;

. .

.)

or

such arithmetical trivia as two plus two

equals four. In

1900 Sikola Tesla declared

that he had received radio signals from in-

telligent beings

on llars. Twenty-one years

later Guglielmo

Marconi made a similar

announcement. Spiritualists too were

~II

frequent contact with minds on the red

planet. The

most remarkable was Hdl6ne

Smith,

a

Swiss medium, whose strange story

is told in the book

From

India

to

the Planet

Mars:

,4

Study of

a

Case oj'Somrztlnzbulisrn

with

Glossol(lliu

(1900)

by

the Swiss psy-

chologist

Thkodore Flournoy. In her

trances

Heli.ne seeined to be under hlartiail

control, speaking and writing a complex

Xlartian language, complete with its own

alphabet.

(On Hekne, and other mediunls

who claimed hlartian contacts, see also

Chapter

8,

"From Kensington to the Planet

liars,"

in Harry Price's

Cor!fessiotas

of

u

Ghost-Hu~ter;

Sew York: Putnan~, 1936.)

Sow that

we are close to landing ex-

ploratory robots on

llars and are expecting

to find, at the most, only a low-grade vege-

tation, interest in extraterrestrial

cornmuni-

cation has shifted to planets in other solar

systems. In 1960 Project

Ozma failed to

detect any radio

messages fro111 outer space

after several months of listening near the

frequency at which free hydrogen radiates.

(For various reasons this frequency, with

its

\vavelength of 21 centimeters, seeins to

be the ideal frequency

for

interstellar

communication.) Severtheless, interest

both in

sending and in searching for such

messages continues, and much abstruse

work is

being done on the best methods of

exchanging

illforination with an alien cul-

ture once contact is established. It is a

fascinating problem, almost the exact op-

posite of devising wartime codes. The pur-

pose of

a

code is to transmit information

in such a

wa~7 as to make it as difficult as

possible for anyone not knowing the key

to

understa~ld the message. The purpose

of an interstellar code is to communicate

with minds that know nothing of our lan-

guage, and in such

a

way as to make it as

ecr.yy

as possible for them to understand.

Many

of

the papers

in

Interstellur

Com-

nltlrzication,

edited by

A.

C,.

LV.

Cameron

(1963), are concerned with this task. ,411

experts agree that messages had best start

with simple arithmetic.

One assumes that

units can be counted by any type of intel-

ligent creature, and that arithmetical laws

Gardner M. Sixth Book of Mathematical Diversions from Scientific American

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