48 Normat 53:1, 48 (2005)

Summary in English

Audun Holme, Editorial (Norwe-

gian). The new editor explains his vi-

sions for Normat.

Harald Hanche-Olsen, On the area

of curves (Norwegian). A brief history

of spaceﬁlling curves. The article warms

up by showing that a curve of ﬁnite

length has zero area, then notes how

Cantor discove red that the unit interval

can be put in a one-to-one correspon-

dence with the unit square – albeit not

in a continuous way. Peano found a con-

tinous mapping of the unit interval onto

the unit square, and Hilbert simpliﬁed

his construction. Finally, the author e x-

plains Osgo od’s construction of a simple

curve – a Jordan arc – of area arbitrarily

close to 1 lying inside the unit square.

Bengt Ulin, Pappus – a juggler of pro-

portions (Swedish). The competence of

a good mathematics teacher includes

the ability of showing instructive and

fascinating history. The ancient mathe-

matician Pappus should be given much

more attention. As a geometer, he was

as ingenious as Archimedes . The aim

with the article is to show how Pappus,

thanks to brilliance and patience, beau-

tifully extended the arbelos theorems by

Archimedes , essentially using analogies

and similarity.

Dag Normann, On possible and ap-

parently impossible programming tasks

(Norwegian). The author discuss the

P = NP problem. Examples of NP

problems discussed are “the traveling

salesman problem”, “the partition prob-

lem”, and a problem related to the

“Mineswe eper” game. Examples of P

problems discussed are the solvability of

a ﬁnite set of linear equations and the

existence of Euler circuits. The paper is

non-technical.

Audun Holme, Lejeune Dirichlet

(Norwegian). A biographical sketch of

Lejeune Dirichlet.

Marius Overholt, Counting in num-

ber theory (Norwegian). The author

gives a brief exposition of the most el-

ementary aspects of the divisor prob-

lem of Dirichlet. This problem concerns

the asymptotic behaviour of the average

number of divisors of the integers from

1 to n as n tends to inﬁnity. In 1849

Dirichlet gave a surprisingly good esti-

mate. In this article a weaker version of

his result is proved by an argument us-

ing the minimum of prerequisites.

Oddvar Iden, Construction of the reg-

ular heptadecagon (Norwegian).

C. F. Gauss showed that this construc-

tion problem leads to four quadratic

equations. The author points out that

by solving them geometrically (instead

of algebraically) one is guided step by

step to a c onstruction.

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