Sierpinski (<1K) PRELIMINARIES

Sierpinski (<1K) Introduction

Publication Issues

Shareware Copyright and Donations
    Copyright in the material on this website is retained. Material is "published" here on a shareware basis. All moral rights of authorship are retained. If you derive benefit from the material here please consider making a donation via PayPal.   

    The withheld material on this site (listed in yellow on the contents page), is no longer available for purchase. Interested publishers should contact the author.

    This work is intended primarily for programmers modelling physical processes. It is intended as a reference and fast tutorial.
    Beginner tutorials in much of the material covered here can be found elsewhere.
    Though advanced mathematics is not required to use the techniques described, some familiarity with standard mathematical notations is assumed. These notations are described in the glossary. Algorithms are described rather than presentated in a particular computer language.
    In this work I eshew the mathematical traditions of theorems, lemmas, numbered equations, and references for the sake of brevity and flow. These can be found in fuller more formal works. Sketch proofs are provided for some important results. Fleshing them out can be considered excercises for those so inclined. The intention here is to foster understanding and rapidly facilitate use in the readers field of interest. Hopefully, this will appeal to the "what can I do with it?" programmer attitude. Mathematics can be truly beautiful, and it can create great beauty. It remains the ultimate toolkit.

    Please note that i am  not the Ian Bell who edited "Maths for the Building Trades" [1997 - Longman]. I  am the Ian Bell who wrote "Elite".

Browser Issues

    These pages require the SYMBOL font and use the old <FONT FACE=SYMBOL> method of accessing the symbols font not supported by Netscape 6 or some installations of Opera 6. I have chosen to retain this method in preference to modern unicode methods since the latter are still only partially supported by modern browsers and fail completely under legacy browsers such as Nestcape 4.5. My recomendation for viewing this work was Netscape 3 or 4.5 but I am now revising the Multivectors section for compatability with Firefox (in Windows).
    Accordingly the mathematial symbols used in pages may not be appear correctly in Netscape 6+ or Opera 6+, but will function fine under earlier versions of these browsers.
    If a capital sigma summation symbol appears here S then these symbols should appear OK in your browser. If an S or a square appears instead, they will not. Some browsers, notably MSIE, display most of the symbol font but fail to display or print the up arrow symbol [↑] which is used repeatedly in these documents.
[  If a sigma and nabla (triangle) appear here: ∑ ∇ then your browser supports the unicode method . ]
    The Jester and Arial fonts are used for additional clarity, but are not strictly required.

    Legacy browsers are available for free download from I recommend installing Netscape3.04 as a small fast safe browser for viewing and printing these and other symbol-font using HTML files when Firefox fails.  

Sub and Superscripts
    These pages make frequent use of nested sub and superscripts such as vip2 and ex2 . Some browsers, notably MSIE 5 , may partially "flatten" or "collapse" such subscripts. Others will not reduce sizes appropriately. If the symbols v , i and p2 are not clearly distinguished in the example then much of the contents will become confusingly malformated.  The third level of subscript (the final 2 in vip2) is seldom required.

Background Image
    If you have difficulties viewing these files due to the background images    click here for guidance.

Printing problems
    If you have difficulties printing out these files,   click here for guidence.
    Note that Netscape's Print Preview may fail to display the symbol font correctly. This does not necessarily mean that the symbols will not appear OK in the hardcopy. It is consequently worth printing a single test page rather than relying on preview.

Glossary   Contents   Author
Copyright (c) Ian C G Bell 1998
Web Source: or
18 Nov 2006.