This
Interactive Web Book explores the following question. Given
n random numbers chosen from the unit interval [0,1], what is the expected
value of the kth shortest segment so determined?
For example, if you select one number in the interval [0,1] it divides that
interval into two parts. One is larger and the other smaller (or they are
equal). What do you think will be the average (expected) length of the shorter
segment? The surprising answer is 1/4. And if you choose two numbers in the
interval, dividing it into 3 parts, what should you expect the shortest length
to be on average? You guessed it: 1/9.
To
see what is going on here, we translate this problem from probability to geometry
and show how a certain extension to higher dimensions of the notion of trapezoid
may be used to calculate certain integrals of functions defined on simplices
and depending on the barycentric coordinates of the points. We will explain
what "barycentric coordinates" are a bit later.
As
always, the emphasis is on visualization through experimentation. The reader
may do the suggested experiment and see for herself the strange distribution
of of expected lengths. Then the reader may construct triangles in the plane,
and examine the properties of the associated first barycentric subdivision
of the triangle (simplex).
These
experiments lead naturally to the questions: how do you graph and then integrate
certain functions on simplices defined in terms of barycentric coordinates.
The surpisingly simple answer in our case is delivered by geometry in an interesting
illustration both of piecewise linear functions,
and multidimensional integration.
Requires
the free Java MathwrightWeb ActiveX Control to read in your Browser.
For
proper viewing, be sure to use Version
2.10 or later, dated May 12, 2003
Download free MathwrightWeb to view
Microworlds in your browser then press
or
Download and extract the Word 2000 version if you wish to print it (33 pages)
Library members, download the free Mathwright32 Reader, then press
For proper viewing, be sure to use Version 2.10 or later, dated May 12, 2003
Interactive Web Book: Odds and Integrals
Author:
James White
This
Interactive Web Book is a demonstration of a possible shape for mathematics
books of the future. It presents its story in the familiar way that a static
text might present it, but with the exception that the pages of the story
"come to life" and offer the reader the opportunity to make and
test hypotheses, to experiment and explore in a visual and interactive way
many of its main constructions and concepts. For many readers, this active
participation in the story can add a dynamic dimension that will help them
visualize certain of its ideas for the first time. Of course, the reader may
download, extract, and print the document in order to read it in the traditional
way as static text offline between experiments, but these interactions are
not ancillary; they are from the beginning an essential part of the narrative.
All
of the documentation is available in the book itself as page-by-page HTML
Help. Most pages are dynamic interaction pages, where you experiment with
the ideas of the story as it unfolds. They offer simulations and command lines
that range through topics in geometry, algebra, and single-variable calculus.
As you come to each page, open the Help, Help for This Page menu to read (or
print) the story, then roll up your sleeves and ... play!
The
mathematical documentation for this story was created fairly easily using
Design Science MathPage™ technology with Microsoft Word 2000™ and
Microsoft Help Compiler. And the mathematical interactions were created with
Mathwright32 Author™. The Odds and Integrals Book may be read in Microsoft
Windows™ (95, 98, Me, 2000, or XP) using Internet Explorer Browser 5.0
or later. In order to read it, you should sure you have MathwrightWeb Version
2.10 (after May 12, 2003, or later).
The
book covers some topics in probability, in multidimensional calculus, at the
level of Calculus 2, in linear algebra and combinatorial topology, and makes
use of geometry and 3 dimensional graphics. It is not
meant to "teach" those traditional topics in tandem with course
instruction, but instead can be used to illustrate and motivate a few of them
in a way that a traditional text cannot do.
To
visit our Interactive Web Books in your browser, it must be able to read
ActiveX controls. Microsoft Internet Explorer 4.0 Browser (or
later) is so equipped. You should check that the Security Settings under
Tools, Internet Options, Security for the Internet, Custom Level
has:
-
"Run ActiveX Controls and Plugins" set either to enable or prompt.
-
"Initialize and Script ActiveX Controls not marked as safe" set either to enable or prompt.
Return to the listing of MathwrightWeb Microworlds
| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
| MathwrightWeb, and Mathwright32 |
