**ERIC Number:**EJ974985

**Record Type:**Journal

**Publication Date:**2012

**Pages:**4

**Abstractor:**ERIC

**Reference Count:**6

**ISBN:**N/A

**ISSN:**ISSN-0045-0685

Diversions: Hilbert and Sierpinski Space-Filling Curves, and beyond

Gough, John

Australian Mathematics Teacher, v68 n2 p30-33 2012

Space-filling curves are related to fractals, in that they have self-similar patterns. Such space-filling curves were originally developed as conceptual mathematical "monsters", counter-examples to Weierstrassian and Reimannian treatments of calculus and continuity. These were curves that were everywhere-connected but nowhere-differentiable (or some similar paradoxical combination of conditions): that is, there were no breaks in the curves, but they were so extremely and discontinuously wiggly that ordinary differentiation did not apply to them. Moreover, they showed that a "line"--specifically a "curve", rather than a "straight line"--could fill two-dimensional space. As early as 1940, the great mathematics popularisers Kasner and Newman discussed the Koch snowflake, the anti-snowflake, and bizarre space-filling "curves" as examples of what Kasner and Newman called "pathological" shapes. Pathological, because the two-dimensional snowflake curve, for example, is contained within a finite area but is itself infinitely long, while the three-dimensional counterpart is a space-filling curve that is infinitely long and completely fills a finite volume. The author offers a few suggestions for materials to read about the great "popularisers" of mathematics.

Descriptors: Geometric Concepts, Calculus, Mathematics Instruction, Mathematics Education, Grade 9, Secondary School Mathematics, Foreign Countries, Mathematics Teachers, Computer Uses in Education

Australian Association of Mathematics Teachers (AAMT). GPO Box 1729, Adelaide 5001, South Australia. Tel: +61-8-8363-0288; Fax: +61-8-8362-9288; e-mail: office@aamt.edu.au; Web site: http://www.aamt.edu.au

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Grade 9; Secondary Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A

**Identifiers - Location:**Australia