tial arrangements. (It was argued that since no single

human being could possibly verify that the computer

had completed this work correctly, then the outcome

produced is not a valid proof.) Most mathematicians

today accept the validity of Appel and Haken’s work

and consider the famous four-color problem solved.

(The search for a simple and elegant proof easily

checked by human hand, however, continues.)

It is interesting to note that Heawood’s conjecture

actually fails for the single case that remains: it is

known that six colors suffice to color any map drawn

on a Klein bottle (g= 1 ), not seven, as predicted by

the formula.

See also

FLOOR

/

CEILING

/

FRACTIONAL PART

FUNCTIONS

.

Fourier, Jean Baptiste Joseph (1768–1830) French

Analysis, Engineering Born on March 21, 1768,

French scholar Joseph Fourier is remembered in mathe-

matics for his fundamental contributions to the theory

of heat conduction and his study of trigonometric

series. His groundbreaking advances in these topics

appear in his famous 1822 treatise Théorie analytique

de la chaleur (Analytic theory of heat). Today, F

OURIER

SERIES

play a fundamental role in the study of physics

and engineering, as well as in the development of theo-

retical mathematics.

Placed in a military school as an orphan at age 10,

Fourier exhibited a strong interest in mathematics at

this early age. He taught at the same military school for

four years before entering the newly established

teacher-training school L’École Normale (later renamed

L’École Polytechnique) in 1794. There Fourier received

instruction in mathematics from J

OSEPH

-L

OUIS

L

AGRANGE

(1736–1813) and excelled in all his studies.

In 1797, when Lagrange stepped down as chair of

analysis and mechanics, Fourier was appointed the new

department chair.

This position was short-lived, however, for the fol-

lowing year Fourier was assigned to Napoleon’s army

in the invasion of Egypt. He was charged to oversee a

variety of archeological and scientific investigations,

and it was during this period of his life, while stationed

in the Egyptian desert, that Fourier developed a fasci-

nation for the mathematics of heat transfer.

Taking a highly original approach, Fourier ana-

lyzed conduction by representing complicated oscillat-

ing quantities as sums of simpler components. This led

him to his theory of trigonometric series. In 1807

Fourier detailed the results of his work in the paper

“On the Propagation of Heat in Solid Bodies,” but the

mathematicians of the day, including Lagrange, were

not convinced of the validity of his approach. Fourier

rewrote the paper in 1811, and even received a prize

for its content, but still received criticism from the

mathematics community for its lack of rigor and lack

of generality. For another 10 years Fourier worked to

establish the mathematical foundations of his work

and detail the reasoning behind his methods. Finally,

in 1822, after the publication of his famous treatise,

all criticisms of his mathematics were settled, and

Fourier’s approach was accepted as valid and funda-

mentally important.

Fourier was elected to the powerful position of sec-

retary to the Académie of Sciences that same year, and

in 1827 he was honored with membership to the

R

OYAL

S

OCIETY

of London.

It has been conjectured that Fourier, ironically, may

have contracted the exotic illness myxedema while in

Egypt, which, among several symptoms, induces an

extreme sensitivity to cold. It is said that Fourier kept

his living quarters almost unbearably hot and always

bundled himself in many layers of clothing. Fourier

died of a heart attack on May 16, 1830.

Fourier series have found numerous applications to

physics and engineering. Practically every branch of sci-

ence that contends with the transfer of energy through

waves (such as acoustics, seismic studies, wireless com-

munications, spectroscopy) utilizes Fourier’s methods.

In a more abstract setting, Fourier series are now seen

as just one of infinitely many possible approaches to

decomposing elements of a

VECTOR SPACE

of functions

in terms of basis elements. This shift in perspective

has led to new insights into the study of quantum

mechanics.

Fourier series When a musician plays two strings on

a violin simultaneously, just one sound wave reaches

our ears—the combined effect of the two notes.

Nonetheless, the human brain is able to decode the

information it receives to “hear” two distinct notes

being played. That is, the human brain is able to recog-

nize complicated sound waves as sums of simpler basic

sound waves.

Fourier series 201