248 helix

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–

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4

1

–

3

1

–

2

=

n

1

21+

Note that in the first leg of the journey, the ant

covers 1/12th of the length of the band, and that this

proportion remains the same as the band is stretched.

During the second leg of the journey, the ant covers

now only 1/24th of the length of the band (1 in. of 24

in.), 1/36th of the length during the third length, and so

on. Thus after nlegs of the journey, the ant covers the

fraction of the band. The

ant reaches the finish only if Hnever surpasses the

value 12. As the harmonic series diverges, this must

indeed be the case.

Although the harmonic series diverges, the

ALTER

-

NATING SERIES

test from the study of

CONVERGENT SERIES

shows that the alternating harmonic series, given by

1 – + – + – +…

converges. We can use Euler’s approximation formula

to find its value. First note:

where the error is no larger than 2/n and so converges

to zero as ngrows. Consequently, the even partial

sums of the series approach the value In(2). The sum

of an odd number of terms equals

ln2 + error + and so too

approaches ln2 as nbecomes large. Thus we have:

1 – + – + – + … = ln(2)

This example is often used to illustrate the difference

between conditional and

ABSOLUTE CONVERGENCE

of a

series. In that setting, it can also be used to provide an

amusing “proof” that 1 equals 2.

See also M

ERCATOR

’

S EXPANSION

.

helix A spiral-shaped curve sitting in three-dimensional

space is called a helix. The name is the Greek word for

a “spiral” or a “twist.”

A cylindrical helix lies on a cylinder and cuts

across straight lines drawn along the length of the

cylinder at a constant angle α. A spiral staircase and

the thread on a straight screw are examples of cylindri-

cal helices. A conical helix is a spiral curve on a

CONE

,

and a spherical helix is a spiral on a

SPHERE

that cuts

lines of longitude at a constant angle.

A cylindrical helix has

PARAMETRIC EQUATIONS

: x

= acost, y = asint, and z= bt, where aand bare con-

stants, and tis the parameter. A conical helix is given

by: x= aetcost, y = aetsint, and z= et.

Heron of Alexandria (ca.10–75

C

.

E

.) Greek Geome-

try, Number theory, Physics, Engineering Sometimes

called Hero, Heron of Alexandria is remembered in

mathematics for his three-volume text Metrica, redis-

covered in 1896. The work discusses and develops in

great detail the principles of geometry, number, and

numerical approximation. It also contains the earliest

known proof of the famous formula that bears Heron’s

name. Outside of mathematics, Heron is best known

for his contributions to mechanics and fluid mechanics.

Demonstrating a wide range of scientific interests,

Heron wrote studies in optics, pneumatics (the study

and use of gas and fluid pressures), astronomy, survey-

ing techniques, and planar and solid geometry, but it is

the work presented in Metrica that proves his genius as

a mathematical intellect. Book I of this famous piece

computes the areas of triangles, quadrilaterals, and reg-

ular polygons, as well as the surface areas of cones,

prisms, spheres, and other three-dimensional shapes.

His famous formula for the area of a triangle solely in

terms of its side-lengths is presented in this section,

along with a general method for computing the square

root of a number to any prescribed degree of accuracy.

(This procedure is today called H

ERON

’

S METHOD

.

Book II of Metrica derives formulae for the volume

of each P

LATONIC SOLID

, as well as the volumes of

cones and spherical segments, and Book III extends

E

UCLID

’s study of geometric division. Heron also pre-

sents a method for determining the cube root of a num-

ber in this third volume.

Heron also wrote a number of important treatises

on mechanics, many of which survive today. His text

Pneumatica represents a careful (but, in places, inaccu-

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2