θ

–

2

340 mode

Möbius band

can also obtain a Klein bottle by “folding” a Möbius

band in half along the central line parallel to its edge and

gluing together the points of the edge that meet. (If one

does this for an ordinary band of paper containing no

half twists, the resulting surface is a

TORUS

.) Unfortu-

nately these constructions cannot be fully completed in

three-dimensional space, and one must make use of the

fourth dimension to obtain sufficient maneuverability.

In a C

ARTESIAN COORDINATE

system, if one rotates

a line segment in the yz-plane about the z-axis, the

resulting surface of revolution is a band with no half-

twists. If that line segment were to rotate about its

MID

-

POINT

180°during the course of being rotated about the

z-axis, then the resulting surface is a Möbius band. For

convenience, suppose the line segment is of length 2 and

is initially positioned in the yz-plane parallel to the z-

axis with midpoint at distance position 2 along the y-

axis. Each point on the line segment is determined by a

parameter v, between –1 and 1, with v= 0 correspond-

ing to the midpoint of the segment. Suppose that when

the segment has turned an angle θabout the z-axis, the

segment has turned an angle about its midpoint.

Then a careful analysis of the positions of points along

this segment as they are rotated about the z-axis shows

that the

PARAMETRIC EQUATIONS

of the resulting Möbius

band are given by:

Mathematicians have shown that, for any map of

regions drawn on a Möbius band, at most six colors

would ever be needed to paint the design so that no two

regions sharing a boundary are painted the same color.

The Möbius band was independently discovered

by German mathematician Johann Listing (1808–88)

and German scholar A

UGUST

F

ERDINAND

M

ÖBIUS

(1790–1868). The one-sided nature of the band was

later exploited by the B.F. Goodrich Company in their

design of Möbius-like conveyor belts. By spreading the

“wear and tear” on both sides of a strip, these belts

lasted twice as long as conventional belts.

See also

DIMENSION

;

FOUR

-

COLOR THEOREM

;

SOLID

OF REVOLUTION

.

mode See

STATISTICS

:

DESCRIPTIVE

.

modular arithmetic The numerals on the face of a

clock provide a model for an unusual mathematical

system called “clock math” or “arithmetic mod 12.”

One thinks as follows: if it is currently 3:00, then 8

hours later it will be 11:00. We write 3 + 8 = 11, noting

nothing unusual here. However, waiting 6 hours from

10:00, say, gives the equation 10 + 6 = 4, for the time

at the end of that wait will be 4:00. Following this new

interpretation for addition, clock math gives, for exam-

ple, 4 + 11 = 3, 8 + 2 = 10, and 7 + 7 = 2.

It is convenient to call the number 12 “zero.”

(After all, in clock math, adding 12 hours to any time

does not change the time registered on the clock and so

has no effect in this system.) The number 13 is

regarded the same as 1, (the 13th hour on a clock lies

at the same position as the first hour), the number 14 is

2, and so forth. In general, clock math replaces any

number with its excess over a multiple of 12. For

example, 26 is two more than a multiple of 12, and so

26 is equivalent to 2. We write 26 ≡2 (mod 12). Simi-

larly, 29 ≡5 (mod 12), 43 ≡7 (mod 12), and 72 ≡0

(mod 12). The number –2 is 10 more than a multiple of

12 and so –2 ≡10 (mod 12). The symbol ≡is called

CONGRUENCE

.

One can perform multiplication in clock math. We

have, for instance, that 3 ×7, normally 21, equals 9 in

clock math: 3 ×7 = 9. (This can also be realized in

terms of repeated addition: 3 ×7 = 7 + 7 + 7 = 2 + 7

= 9.) In the same way, 2 ×4 = 8, 4 ×5 = 8, and 6 ×6 =

0 in clock math. The following table shows all prod-

ucts in this system.