The Mathematica® Journal
Making MathWorld
Eric W. Weisstein
MathWorld (mathworld.wolfram.com) is the internet’s most popular
mathematical resource. Its extensive and detailed entries usually find their
way into the top three (if not top one) results returned by a Google search
for mathematics terms. Mathematica has long been instrumental in the
authoring of MathWorld, which has been hosted by Wolfram Research
since 1999. As a result of new work partially funded by a grant from the
National Science Digital Library program of the National Science Foundation, MathWorld’s contents are now written entirely in notebooks that are
converted to web pages by Mathematica itself. In this article, the processes
that bring MathWorld from keyboard to the web will be discussed, focusing especially on the Mathematica-based tools that make this transformation possible. In addition, a number of useful, new interactive features
added to MathWorld as a part of the digital library work will be discussed.
‡ What Is MathWorld?
MathWorld, formerly known as Eric Weisstein’s World of Mathematics, is an online
encyclopedia of mathematics provided by Wolfram Research as a free resource to
the world mathematics community. It is not only an accessible and up-to-date
collection of mathematical knowledge, but also serves as an extensive database of
literature references and links to mathematics on the web. In addition to its
thousands of entries, MathWorld is amply illustrated with thousands of figures
and diagrams, interactive applets for visualizing three-dimensional geometric
objects, and a collection of embedded webMathematica-based demonstrations.
The main page of the site contains links to MathWorld Headline News, which
provides in-depth coverage of breaking mathematics news. Finally, an online
classroom section has recently been added, providing mathematical entry points
that are accessible to students and educators.
According to reliable sources (in particular the all-knowing and all-powerful
PageRank™ algorithm on Google™: www.google.com/Top/Science/Math),
MathWorld is the most popular math website on the internet. This is also
reflected by the fact that doing a Google search for arbitrary math terms generally returns MathWorld pages near the top of the list, most commonly in the
number one spot. Feel free to try this experiment yourself; suggested starting
points are “eigenvalue,” “square matrix,” “Khinchin’s constant,” “cylindrical
algebraic decomposition,” or “Fourier transform.”
MathWorld receives hundreds of thousands of page hits each day. Analysis of
the site’s readership shows that it originates from a broad and diverse set of
sources, with technical companies and universities representing the largest share.
As a result of its large audience, MathWorld accounts for the majority of web
traffic at Wolfram Research. In addition, more Mathematica notebooks are
downloaded from MathWorld than from any other website.
The Mathematica Journal 10:3 © 2007 Wolfram Media, Inc.
MathWorld
receives hundreds of thousands of page hits each day. Analysis475
of
Making MathWorld
the site’s readership shows that it originates from a broad and diverse set of
sources, with technical companies and universities representing the largest share.
As a result of its large audience, MathWorld accounts for the majority of web
traffic at Wolfram Research. In addition, more Mathematica notebooks are
downloaded from MathWorld than from any other website.
While many readers may realize that MathWorld is a vast collection of mathematical material, they may not fully appreciate its true extent. As of September 2006,
MathWorld is equivalent to a 4,251-page book (in 9-point type and on 8.5µ11"
paper) containing more than 12,500 entries, 10,000 graphics, and 100,000 crosslinks. Unlike a book, however, it also contains a variety of interactive components. These include more than 400 entries with interactive Java applets for solid
geometry (topics/LiveGraphics3DApplets.html), 100 entries with animated GIFs
(topics/AnimatedGIFs.html), and, more recently, nearly 100 entries with interactive webMathematica examples (topics/webMathematicaExamples.html).
In addition to interactivity, MathWorld contains a large number of computational
resources that may be helpful to Mathematica users. There are nearly 4,000
downloadable Mathematica notebooks on the site, each of which contains code
implementing algorithms, illustrations, and computations related to a given
entry. These sample notebooks can be downloaded by clicking the “Download
Mathematica Notebook” link at the top of a relevant page, as illustrated in
Figure 1 for Reversal.html. In this case, the notebook contains some simple code
for computing the reversals of numbers (i.e., numbers formed by reversing the
decimal digits of a given number and concatenating), defining the functions
Reversal and PalindromicQ.
Figure 1. Navigation elements for Reversal entry.
In[1]:=
In[2]:=
Reversaln_Integer, b_: 10 :
FromDigitsReverseIntegerDigitsn, b, b
PalindromicQn_ : Modulel  IntegerDigitsn, l  Reversel
This code can be used, for example, to find the nontrivial (i.e., nonpalindromic)
numbers less that 106 whose reversals are integer multiples of themselves:
In[3]:=
Out[3]=
SelectRange10^ 6,
 PalindromicQ# && ModReversal#, #  0 &  Timing
32.99 Second, 1089, 2178, 10989, 21978, 109989, 219978
(These are the first few terms of sequence A008918 in “The On-line Encyclopedia of Integer Sequences” by Neil Sloane: www.research.att.com/~njas/
sequences/A008918.)
In addition to downloadable notebooks, more than 50 Mathematica packages
(available from the Mathematica Library Archive at library.wolfram.com/
infocenter/MathSource/4775) provide functionality used to create MathWorld
content.
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Eric W. Weisstein
‡ Authoring MathWorld Before Mathematica
Prior to July 2005, MathWorld was authored in TEX, a language that is widely
used for typesetting mathematics but is ill-suited for translation into modern
markup languages such as HTML and XML. As a result, it is generally difficult
to convert TEX documents into a format suitable for publication on the web [1].
The MathWorld build system accomplished the TEX-to-HTML conversion
using a complicated multistep translation process that utilized: (1) a customized
version of LATEX2HTML; (2) external image-rasterization libraries; (3) perl
database table construction and element extraction; and (4) multiple passes of
post-processing on the resulting files. This meant that the build system relied on
a large number of moving parts, several software components, and multiple steps.
As a result, it was relatively fragile and hence difficult to maintain—let alone
extend. In addition, the equations produced by the process were limited to static
GIF images (essentially, pictures of the original equations displayed in pixel
form).
Pictures of equations have many severe limitations. They cannot be enlarged,
rescaled, linebroken, searched, crawled, or fed into speech synthesis systems to
provide accessibility to the visually impaired. In addition, because images contain
no semantic information, equations so represented cannot be extracted and
plugged into a software system such as Mathematica in order to perform computations. A possible solution to these limitations would be to use an XML-based
representation for encoding equations that is capable of being directly rendered
in browsers. Happily, such a representation exists for mathematics and is known
as MathML. MathML has a W3C-recommended standard (www.w3.org/Math),
native rendering support in Mozilla and in other browsers via several commercial
plug-in products, and built-in export and import capabilities by mathematical
software systems such as Mathematica. Unfortunately, widespread adoption of
MathML has thus far been hampered by a number of problems that make it
difficult to use as a display technology within an arbitrary browser running on an
arbitrary operating system.
A further drawback of TEX-based authoring was that while results were derived,
verified, and visualized using Mathematica, all derived equations required manual
(or semi-manual) transcription into TEX as a result of the fact that textual content, graphics, and executable Mathematica code were all maintained separately.
‡ Authoring MathWorld with Mathematica
Between 2002 and 2005, the MathWorld build system was redesigned and rebuilt
from the ground up. This work was undertaken with partial support from a
National Science Foundation Digital Library grant to add new interactive
components to the site that would be of value to the hundreds of thousands of
students, teachers, and researchers who routinely visit MathWorld. In order to
accomplish this goal, we created a modern and robust system for building MathWorld that both allows simple authoring of mathematical content and permits
easy inclusion of interactive components. In particular, the new system was
designed to allow incorporating interactive calculators and plotters, the ability to
map MathWorld’s existing subject classification system into the well-established
Mathematical Subject Classification (MSC) scheme used by mathematicians, the
addition
of aJournal
rich10:3
set© 2007
of metadata
onInc.each MathWorld page to allow its contents
The Mathematica
Wolfram Media,
to be described and classified, the creation of a new didactic layer for navigation
of its content, and the capability to generate pages in multiple formats, especially
MathML.
components to the site that would be of value to the hundreds of thousands of
students, teachers, and researchers who routinely visit MathWorld. In order to
accomplish
this goal, we created a modern and robust system for building MathMaking MathWorld
477
World that both allows simple authoring of mathematical content and permits
easy inclusion of interactive components. In particular, the new system was
designed to allow incorporating interactive calculators and plotters, the ability to
map MathWorld’s existing subject classification system into the well-established
Mathematical Subject Classification (MSC) scheme used by mathematicians, the
addition of a rich set of metadata on each MathWorld page to allow its contents
to be described and classified, the creation of a new didactic layer for navigation
of its content, and the capability to generate pages in multiple formats, especially
MathML.
The powerful capabilities of Mathematica provided an ideal system for achieving
these goals. In particular, Mathematica has extensive built-in knowledge of
HTML, XHTML, and MathML, as well as the ability to perform complicated
sets of pattern-based transformations on structured documents. In addition to its
utility as a system for building websites, Mathematica is also a powerful authoring
environment, symbolic/numerical computation engine, and has extensive
import/export capabilities. It offers WYSIWYG for easier authoring and greater
efficiency, access to Mathematica notebook/palette programming as tools for
authoring, and provides its own markup language standard (the notebook) to
allow content to be easily authored and incorporated. It also automatically handles other issues that can be difficult for traditional authoring environments, for
example, line breaking and equation numbering. And because of Mathematica’s
ability to take its own structured document format and convert it to almost any
desired format, the same core build system can be used in a modular way to generate content in multiple formats, in this case (X)HTML and MathML.
‡ Getting MathWorld into Mathematica
Before creating a Mathematica-based build system, it was first necessary to
undertake a one-time conversion of the MathWorld source documents from TEX
into Mathematica notebooks. Such conversions are challenging, especially since
TEX and LATEX are presentation- rather than semantics-based and hence
accurately describe only the intended positions of blobs of ink on the printed (or
electronic) page rather than the actual meaning of those symbols. Because of its
ubiquity as a typesetting language for mathematics, a fair amount of effort has
been devoted to developing translation programs from TEX to other formats. In
particular, a number of systems have been built that purport to translate TEX
into MathML and Mathematica. In principle, translation from one box-based
language to another is a straightforward process. Unfortunately, the practice is
more often than not fraught with difficulties as soon as nontrivial cases are
encountered, which inevitably happens almost immediately. In fact, after investigating a fairly exhaustive list of existing translators, we determined that none
were flexible or reliable enough to handle the extensive set of source documents
comprising MathWorld. It was therefore necessary to implement our own
TEX-to-notebook conversion.
Fortunately, an internal software tool known as tex2nb had previously been
developed at Wolfram Research that was capable of translating a subset of TEX
into notebooks. While this tool had been used internally for some time, it was
not under active development and required a large number of extensions, modifications, and changes to be capable of (1) parsing the entire contents of MathWorld’s source documents and (2) producing clean notebooks that were amenable to subsequent programmatic parsing. In particular, the tex2nb conversion
had to be done carefully in order to preserve
typesetting
structures,
to tag
The Mathematica
Journal 10:3 ©
2007 Wolfram Media,
Inc.
information that could not be used directly, and to provide highly structured and
uniform notebooks. Happily, we were able to eventually overcome these obstacles. It thus became possible to take the entire set of TEX source documents
Fortunately, an internal software tool known as tex2nb had previously been
developed
at Wolfram Research that was capable of translating a Eric
subset
of TEX
478
W. Weisstein
into notebooks. While this tool had been used internally for some time, it was
not under active development and required a large number of extensions, modifications, and changes to be capable of (1) parsing the entire contents of MathWorld’s source documents and (2) producing clean notebooks that were amenable to subsequent programmatic parsing. In particular, the tex2nb conversion
had to be done carefully in order to preserve typesetting structures, to tag
information that could not be used directly, and to provide highly structured and
uniform notebooks. Happily, we were able to eventually overcome these obstacles. It thus became possible to take the entire set of TEX source documents
(equivalent to a printed book more than 4000 pages long) and produce clean,
machine-readable notebooks in an entirely automated fashion.
Mathematica users may be interested to learn that the inner workings of this tool
have subsequently been integrated into Mathematica itself. In fact,
Mathematica 5.1 and later do a quite passable job of importing vanilla TEX (as
well as a fair bit of vanilla LATEX) directly. To illustrate this, consider the
following example, which defines one of the seven so-called “mock theta functions” of order 3. (For more details on these functions, see the MathWorld entry
MockThetaFunction.html.)
Define one of the mock theta functions of order 3.
In[4]:=
expr  q, s 
Sumq^2 n n  1  Product1  q^2 k  1 ^2, k, 0, n, n, 0, s
q2 n 1n
q, s   

n
k0 1  q12 k 2
s
Out[4]=
n0
This expression can be converted into a LATEX snippet using the command
TeXForm.
Show this expression in TeXForm.
In[5]:=
tex  TeXFormexpr
Out[5]//TeXForm=
\omega (q,s)=\sum _{n=0}^s \frac{q^{2 n (n+1)}}{\prod _{k=0}^n \left(1-q^{2
k+1}\right)^2}
Converting this to a string and surrounding it with the signature “double dollar
signs” that delimit a displayed equation in TEX, then importing the result into
Mathematica by placing it in a notebook, shows that Mathematica does a nice job
of directly importing the LATEX snippet.
Import the resulting TEX into a notebook (Figure 2).
In[6]:=
Out[6]=
NotebookPutImportString"$$"  ToStringtex  "$$", "TeX",
NotebookCreate"WindowTitle"  "MockThetaFunction"
NotebookObjectMockThetaFunction
The Mathematica Journal 10:3 © 2007 Wolfram Media, Inc.
Making MathWorld
479
Figure 2. TEX expression for the mock theta function imported directly into a Mathematica
notebook.
‡ The Result
After running through the translation process from TEX to notebook, here is the
source document for a “typical” entry, in this case Abundance.html. As can be
seen, this notebook incorporates metadata tagging, textual content, typeset
mathematics, tabular material, and references, all together in one conveniently
annotated notebook.
Abundance
Metadata
Subjects
Mathematics:Number Theory:Special Numbers:Digit-Related Numbers
Mathematics:Foundations of Mathematics:Mathematical Problems:Unsolved Problems
History
2005-06-30
Downloads
NumberTheory:Abundance
Entry
The abundance of a number n, sometimes also called the abundancy (a term which in
this work, is reserved for a different but related quantity), is the quantity
AHnL ª sHnL - 2 n,
(1)
where sHnL is the divisor function. The abundances of n  1, 2, … are -1, -1, -2,
-1, -4, 0, -6, -1, -5, -2, -10, 4, -12, -4, -6, -1, … (Sloane’s A033880).
The following table lists special classifications given to a number n based on the
value of AHnL.
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480
Eric W. Weisstein
AHnL class
<0
-1
0
1
>0
Sloane
list of n
deficient number
A005100 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14,
15, 16, 17, …
almost perfect number A000079 1, 2, 4, 8, 16, 32, 64, 128, 256, 512,
1024, 2048, …
perfect number
A000396 6, 28, 496, 8128, …
quasiperfect number
none known
abundant number
A005101 12, 18, 20, 24, 30, 36, 40, 42, 48, 54,
56, 60, …
Values of n such that AHnL is odd are given by n  1, 2, 4, 8, 9, 16, 18, 25, 32, …
(Sloane’s A028982; i.e., the union of nonzero squares and twice the squares). Values
of n such that AHnL is square are given by n  6, 12, 28, 70, 88, 108, 168, …
(Sloane’s A109510).
Kravitz has conjectured that no numbers exist whose abundance is a (positive) odd
square (Guy 2004).
SeeAlso
Abundancy, Deficiency
References
Guy, R. K. Unsolved Problems in Number Theory, 3rd ed. New York: SpringerVerlag, 2004.
Sloane, N. J. A. Sequences A000079/M1129, A000396/M4186, A005100/M0514,
A005101/M4825, A028982, A033880, and A109510 in “The On-Line Encyclopedia
of Integer Sequences.”
‡ Building MathWorld with Mathematica
Once the contents of MathWorld are present in carefully tagged and structured
notebooks, the process of website creation can begin. While Mathematica has the
ability to produce web pages directly via Export and File@Save As Special@
HTML, the MathWorld site contains a number of specialized structures that must
be created during the build. For example, MathWorld contains both alphabetical
(mathworld.wolfram.com/letters) and topical (mathworld.wolfram.com/topics)
indices. The indexing information is contained in tagged cells for each entry, but
in addition to the link trails that are displayed on individual entry pages, index
pages for each topic must also be generated. Similarly, MathWorld contains a
“what’s new” page (mathworld.wolfram.com/whatsnew) which lists entries that
have been added or substantially modified. These custom navigation elements
require special treatment in the website building process.
The production of MathWorld web pages from source notebooks is implemented
using a general-purpose Mathematica-based tool known as transmogrify that is
currently under active development by the online documentation group at Wolfram Research. Transmogrify is in effect an XML processing tool for website
The
Mathematica
10:3 © 2007 Wolfram
Media,
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implements
the idea
ofInc.
Extensible Stylesheet Language Translation
(XSLT) in top-level Mathematica code by making use of Mathematica’s XML and
pattern-matching capabilities. Just as XSLT makes single-source documents possible in XML, transmogrify allows conversion of a Mathematica notebook into any
Making MathWorld
481
The production of MathWorld web pages from source notebooks is implemented
using a general-purpose Mathematica-based tool known as transmogrify that is
currently under active development by the online documentation group at Wolfram Research. Transmogrify is in effect an XML processing tool for website
creation that implements the idea of Extensible Stylesheet Language Translation
(XSLT) in top-level Mathematica code by making use of Mathematica’s XML and
pattern-matching capabilities. Just as XSLT makes single-source documents possible in XML, transmogrify allows conversion of a Mathematica notebook into any
type of XML. In fact, prior to being adapted for MathWorld, transmogrify had
already been used to create many popular websites at Wolfram Research, including The Wolfram Functions Site (functions.wolfram.com) and the Wolfram
Mathematica Documentation Center (documents.wolfram.com).
To use transmogrify, we created a set of XML-like templates that described how
various MathWorld notebook structures should be translated. These templates
could then be customized for HTML, XHTML+MathML, Java Server Pages
(JSPs), and so on, making it easy to export to multiple formats using the same
overall system. We were therefore able to develop a master Mathematica program
based on transmogrify that builds the entire MathWorld website, all content
pages, indices and subject trees, and additionally performs such useful functions
as checking link integrity, building custom indices, and so on. This system also
automatically takes care of rasterizing inline and displayed equations into GIFs
using Mathematica’s built-in ability to export its typeset structures to raster
formats, as well as adding equation numbers and other related matter. It is also
clever enough to rebuild only needed parts of the site, making it now possible to
build and push incremental updates of MathWorld several times a day.
‡ New Features in MathWorld
In addition to building the MathWorld site, the transmogrify-based build system
also made it possible to include a number of entirely new features on MathWorld.
These include
Ë Qualified Dublin Core metadata, including MSC headings
Ë interlinking with The Wolfram Functions Site (functions.wolfram.com)
Ë a new MathWorld Classroom for browsing mathematical content on
MathWorld based on learning prerequisites
Ë new interactive webMathematica examples, plotters, and calculators for
many entries
Ë the ability to create mathematical markup language (MathML) versions
of the site
Ë a streamlined comment and contribution system
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Eric W. Weisstein
· Dublin Core Metadata
The Dublin Core Metadata Initiative (dublincore.org) is an open forum engaged
in the development of interoperable online metadata standards that support a
broad range of needs. In particular, qualified Dublin Core provides a rigorous
and specific set of tags that can be used to precisely describe the contents of a
web resource. All of the nearly 13,000 entries on MathWorld now contain Dublin
Core metadata, as illustrated below for the MathWorld entry Sphere.html. As can
be seen, this metadata provides a summary of a page’s content, revision history,
and subject classifications in both the proprietary MathWorld scheme and in
MSC, as well as information about the publisher, language, and so on. Inclusion
of this metadata means that MathWorld content can be easily discovered, harvested, and re-exposed by protocols such as the Open Archives Initiative (OAI),
thus allowing, for example, the type of federated searching provided by the
National Science Digital Library project.
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>Sphere -- from Wolfram MathWorld</title>
<meta http-equiv="content-type" content="text/html; charset=iso-8859-1" />
<meta name="DC.Creator" content="Weisstein, Eric W."/>
<meta name="DC.Description" content="A sphere is defined as the set of all
points in three-dimensional Euclidean space R^3 that are located at a
distance r (the &quot;radius&quot;) from a given point (the
&quot;center&quot;). Twice the radius is called the diameter, and pairs of
points on the sphere on opposite sides of a diameter are called antipodes.
Unfortunately, geometers and topologists adopt incompatible conventions
for the meaning of &quot;n-sphere,&quot; with geometers referring to the
number of coordinates in the underlying space..."/>
<meta name="DC.Date.Modified" scheme="W3CDTF" content="2002-07-17"/>
<meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-04-15"/>
<meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-09-05"/>
<meta name="DC.Date.Modified" scheme="W3CDTF" content="2004-04-21"/>
<meta name="DC.Date.Modified" scheme="W3CDTF" content="2005-02-17"/>
<meta name="DC.Subject" scheme="MathWorld"
content="Mathematics:Geometry:Solid Geometry:Spheres"/>
<meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History
and Terminology:Mathematica Commands"/>
<meta name="DC.Subject" scheme="MathWorld"
content="Mathematics:Recreational Mathematics:Interactive
Entries:LiveGraphics3D Applets"/>
<meta name="DC.Subject" scheme="MSC_2000" content="51M04"/>
<meta name="DC.Rights" content="Copyright 1999-2006 Wolfram Research, Inc.
See mathworld.wolfram.com/about/terms.html for a full terms of use
statement."/>
<meta name="DC.Format" scheme="IMT" content="text/html"/>
<meta name="DC.Identifier" scheme="URI"
content="mathworld.wolfram.com/Sphere.html"/>
<meta name="DC.Language" scheme="RFC3066" content="en"/>
<meta name="DC.Publisher" content="Wolfram Research, Inc."/>
<meta name="DC.Relation.IsPartOf" scheme="URI"
content="mathworld.wolfram.com"/>
<meta
name="DC.Type"
scheme="DCMIType"
content="Text"/>
The
Mathematica
Journal 10:3 © 2007
Wolfram Media, Inc.
...
</head>
See mathworld.wolfram.com/about/terms.html for a full terms of use
statement."/>
Making
MathWorld
<meta name="DC.Format"
scheme="IMT" content="text/html"/>
<meta name="DC.Identifier" scheme="URI"
content="mathworld.wolfram.com/Sphere.html"/>
<meta name="DC.Language" scheme="RFC3066" content="en"/>
<meta name="DC.Publisher" content="Wolfram Research, Inc."/>
<meta name="DC.Relation.IsPartOf" scheme="URI"
content="mathworld.wolfram.com"/>
<meta name="DC.Type" scheme="DCMIType" content="Text"/>
...
</head>
483
Listing 1. Dublin Core metadata for Sphere.html.
· Interlinking with The Wolfram Functions Site
The Wolfram Functions Site is a sister site of MathWorld that consists of nearly
90,000 mathematical identities and more than 10,000 visualizations of the
elementary and special functions of mathematical physics (and, in particular, of
those implemented in Mathematica). As such, it gives many more formulas and
identities satisfied by any given function than MathWorld can. At the same time,
the handbook-style presentation of The Wolfram Functions Site provides less
motivation and background than does the corresponding MathWorld entry. As a
result, it makes a great deal of sense to interlink these two large resources. This
process has now been completed; readers of The Wolfram Functions Site will
note links to MathWorld from its sidebar, while MathWorld readers will notice
special links near the references section of certain entries to corresponding pages
on The Wolfram Functions Site.
· MathWorld Classroom
The MathWorld Classroom is an entirely new part of MathWorld designed to
help students and educators obtain streamlined definitions and didactic information for a select group of entries. Classroom pages indicate in which course and
level of a typical mathematical curriculum a given entry falls, and include educational standards material in addition to examples of and prerequisites for the
given concept. The pages also provide interlinking and cross-navigation between
the main MathWorld entry and the Classroom entry. By analyzing mathematics
curricula, textbooks, and access logs on MathWorld, we were able to compile a set
of approximately 300 core entries which we targeted for inclusion in the Classroom. A concise and accessible definition was then carefully written for each such
entry, and a database containing information about the educational level and
relationships of that concept to others in the Classroom was constructed.
An illustration of a typical Classroom entry is shown in Figure 3. It includes (1) a
cross-link to the full MathWorld entry; (2) a definition; (3) the educational level;
(4) educational standards in which the entry appears; (5) examples of the entry;
(6) other Classroom articles covering similar material; and (7) other Classroom
articles for material that would be encountered in the same course.
The Mathematica Journal 10:3 © 2007 Wolfram Media, Inc.
484
Eric W. Weisstein
Figure 3. MathWorld classroom entry for Quadrilateral.
The Classroom is easily accessible in a number of different ways. A convenient
link on the sidebar of each MathWorld page takes the reader to an overview page
(mathworld.wolfram.com/classroom) listing the names of mathematics courses
that an American student would typically encounter from elementary school on
through graduate course work. Clicking a given course lists all entries in the
Classroom that would normally be encountered in that course, and clicking any
one of these opens a popup window like the one illustrated in Figure 3. Classroom popup windows can also be opened from the corresponding full MathWorld entry. When browsing Quadrilateral.html, for example, a small icon
appears at the top of the entry indicating that the user can “Explore this topic in
the MathWorld Classroom.” Conveniently, the user can easily navigate back and
forth between Classroom entries, which always open in the popup window, and
main MathWorld entries, which always open in the usual browser window, since
each Classroom entry contains a corresponding “Explore this topic in MathThe
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World”
icon Journal
and link.
one of these opens a popup window like the one illustrated in Figure 3. Classroom popup windows can also be opened from the corresponding full MathWorld
entry. When browsing Quadrilateral.html, for example, a small icon
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485
appears at the top of the entry indicating that the user can “Explore this topic in
the MathWorld Classroom.” Conveniently, the user can easily navigate back and
forth between Classroom entries, which always open in the popup window, and
main MathWorld entries, which always open in the usual browser window, since
each Classroom entry contains a corresponding “Explore this topic in MathWorld” icon and link.
The MathWorld Classroom is currently being used extensively by both educators
and students. While there is much additional work that could be done and
additional entries that could be added to the Classroom, the current version
appears to fill a unique void on the internet between sites containing extremely
simple educational resources and those containing very complicated ones. On
MathWorld, the two are now integrated together in a way that is especially useful
to teachers and students alike.
· Interactive webMathematica Examples
While MathWorld contains a huge amount of mathematical content, it is only as
a result of work carried out under the National Science Digital Library grant
that much of this content is now interactive. webMathematica is a web-based version of Mathematica that allows real-time mathematical computations to be incorporated into web-based content. By combining the power of webMathematica
with the ease of website generation provided by MathWorld’s new transmogrifybased build system, it is straightforward to convert static plots into interactive
ones, precomputed tables of values into customized ones computed on-thefly, and so on. MathWorld currently contains 128 interactive webMathematica
examples on 86 separate pages. (For a complete listing, see topics/
webMathematicaExamples.html.) These examples include plotters for functions
on the real line or in the complex plane, but also more complicated examples
such as RiemannSum.html shown in Figure 4, which allows the reader to learn
about Riemann summation by specifying arbitrary functions, endpoints, styles,
and so on.
Due to the ease of creating and incorporating webMathematica examples, we plan
to greatly augment the already large number of such examples over time.
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Eric W. Weisstein
Figure 4. webMathematica interactive example for Riemann Sum entry.
· MathML Versions of Pages
As already mentioned, by making use of Mathematica’s built-in knowledge of
XML and MathML, it is in principle relatively straightforward to use the same
notebooks that currently create HTML pages with GIF images and instead
construct XHTML+MathML versions of the pages using the same build system.
We are currently able to do this, and as a result are presently collaborating with
projects seeking to search arbitrary online mathematics. While MathML versions
of MathWorld pages are very useful and suitable for this purpose, we have thus
far not been able to bring them to a suitably polished state to be able to place
them on the live website. There are a number of reasons for this, some of which
are related to the nature of the MathWorld source documents themselves, but
others of which are due to inherent limitations in the support of the MathML
standard in browsers.
The limitation in MathWorld’s source documents is that, because they began life
as TEX, there is no semantic information accompanying mathematical markup
except that which can be inferred based on the typesetting structures themselves.
The existing formulas therefore can be translated into presentation MathML,
but only with limited semantic information. As it turns out, while the resulting
MathML is sufficient for display, indexing, and searching, only a limited subset
of it is of sufficient quality to allow it to perform computations. There does not
appear to be any shortcut here; making the transition from presentation to
semantic markup requires either rekeying from scratch (an error-prone and
labor-intensive operation) or additional development of software tools for
inferring and tagging ambiguous interpretations (Does the typeset structure
“p Hx - 1L” represent p Hx - 1L, “the quantity pi times the quantity x - 1” or
PrimePi[x], “the prime counting function of the quantity x - 1”?).
The Mathematica Journal 10:3 © 2007 Wolfram Media, Inc.
MathML is sufficient for display, indexing, and searching, only a limited subset
of it is of sufficient quality to allow it to perform computations. There does not
appear
to be any shortcut here; making the transition from presentation487
to
Making MathWorld
semantic markup requires either rekeying from scratch (an error-prone and
labor-intensive operation) or additional development of software tools for
inferring and tagging ambiguous interpretations (Does the typeset structure
“p Hx - 1L” represent p Hx - 1L, “the quantity pi times the quantity x - 1” or
PrimePi[x], “the prime counting function of the quantity x - 1”?).
On the browser side, a number of partial solutions are currently available, but the
fact remains that support for MathML is still very problematic. In particular, a
variety of commercial and public domain tools exist for viewing MathML, but
none of the solutions we investigated were capable of rendering a page of the
complexity of a typical MathWorld entry in a way that would be suitable for
general usage. The situation is significantly complicated by cross-platform and
cross-browser incompatibilities, nontrivial configuration issues that would exceed
the technical abilities of the vast majority of MathWorld readers, font limitations,
and the lack of a viable solution under Mac OS X.
We continue to investigate technical issues surrounding full MathML versions of
MathWorld pages and hope that as some of the technical limitations are overcome by browser and plug-in vendors, it will become possible to view MathWorld
pages in MathML.
· Streamlined Comment and Contribution System
Reader feedback is an indispensable part of MathWorld’s success. In fact, thousands of contributors have been instrumental in building the site into the pre-eminent mathematics resource that it is today. While maintaining the convenient
navigation structures that help make the vast amount of content on MathWorld
easy to navigate, a newly redesigned contribution form accessible through links
at the top of each MathWorld page lets readers leave comments and suggestions
about that specific page. If you have a comment or contribution you'd like to
share, please consider leaving a message!
‡ Conclusions
MathWorld is now authored and built using Mathematica and Mathematica-based
tools. As outlined in this article, MathWorld continues to grow not only in size,
but also with new and useful features. Even more interactive features are on the
way, taking advantage of Mathematica-based technologies such as webMathematica. Finally, I would like to take this opportunity to say what a great pleasure and
honor it has been to have corresponded with so many MathWorld readers and
contributors over the years. Thank you all for your support, readership, comments, contributions, and continued feedback, all of which have enabled MathWorld—which is a labor of love for me—to remain a useful and up-to-date
mathematical resource over the last ten years.
‡ Acknowledgments
The author would like to thank Stephen Wolfram for many useful discussions
and Wolfram Research for its continued support of free public resources that are
of use not only to Mathematica users, but to the world science and mathematics
communities in general. I am grateful to John Renze, who carried out much of
the design and implementation of the new build system in addition to working
extensively on the MathWorld Classroom. Thanks also go out to Chad Slaughter
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Journal 10:3were
© 2007 Wolfram
Media, Inc.
and Bill White, whose TEX-nical and technical
knowledge
invaluable
in
creating the system to translate MathWorld sources into Mathematica notebooks.
Michael Trott was frequently available to lend a small part of his unsurpassed
knowledge of Mathematica programming. This project could not have been
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Eric W. Weisstein
The author would like to thank Stephen Wolfram for many useful discussions
and Wolfram Research for its continued support of free public resources that are
of use not only to Mathematica users, but to the world science and mathematics
communities in general. I am grateful to John Renze, who carried out much of
the design and implementation of the new build system in addition to working
extensively on the MathWorld Classroom. Thanks also go out to Chad Slaughter
and Bill White, whose TEX-nical and technical knowledge were invaluable in
creating the system to translate MathWorld sources into Mathematica notebooks.
Michael Trott was frequently available to lend a small part of his unsurpassed
knowledge of Mathematica programming. This project could not have been
completed without the help of Jean Buck and her uncanny knack for matching
technical needs with the resources required to address them. Thanks also to
Megan Gillette and Jeremy Davis for the elegant graphical designs that make
MathWorld’s appearance worthy of its content. I would especially like to acknowledge Lee Zia and the National Science Foundation National Science Digital
Library program for supporting this work through grant #0226327, and to thank
William Mischo and Tim Cole at the University of Illinois for their expert
knowledge and close collaboration on many aspects of this work.
‡ References
[1] R. J. Fateman and R. Caspi, “Parsing TEX into Mathematics,” (May 6, 2004)
www.cs.berkeley.edu/~fateman/papers/parsing_tex.pdf.
About the Author
Eric W. Weisstein began compiling scientific encyclopedias as a high school student nearly
20 years ago. Born in Bloomington, Indiana in 1969, he studied physics and astronomy at
Cornell University and at Caltech and received his Ph.D. from Caltech in 1996. In 1995,
Weisstein took the vast collection of mathematical facts that he had been accumulating
since his teenage years and began to deploy them on the early internet. These pioneering
efforts at organizing and presenting online content helped define a paradigm that has
subsequently been followed by other large-scale informational projects on the web.
Weisstein joined Wolfram Research in 1999 and unveiled the MathWorld website at
mathworld.wolfram.com later that year. As a Senior Research Fellow at Wolfram
Research, Weisstein has led the development of MathWorld, continuing to expand its
scope and depth and fulfilling his vision for bringing accessible mathematical and
scientific knowledge to the widest possible audience. Weisstein works closely with the
main development teams at Wolfram Research and is a consultant for the CBS television
crime drama NUMB3RS.
Eric W. Weisstein
Senior Research Fellow
Wolfram Research, Inc.
eww@wolfram.com
The Mathematica Journal 10:3 © 2007 Wolfram Media, Inc.