# Page Design Notes from an Undergraduate Student Leader

Welcome to the Math Images wiki!

Introduction Most of the pages on the site are image pages. These pages are organized around a specific image, which is displayed at the top right of the page. Several examples can be accessed from the main portal (http://mathforum.org/mathimages/index.php/Main_Page).

In addition to what you might find there, some representative examples are: Stereographic Projection: http://mathforum.org/mathimages/index.php/Stereographic_Projection Change of Coordinate Systems: http://mathforum.org/mathimages/index.php/Change_of_Coordinate_Systems Four Color Theorem: http://mathforum.org/mathimages/index.php/Four_Color_Theorem Polar Equations: http://mathforum.org/mathimages/index.php/Polar_Equations

In addition to image pages, the Math Images site includes helper pages (http://mathforum.org/mathimages/index.php/Helper_Pages). These pages explain mathematical concepts that don’t lend themselves so directly to a main image but which are required because other image pages on the site draw from them. The present article will focus on image pages rather than helper pages, which have a looser organization depending on the topic and require less explication.

Each image page follows the same basic overall structure.

Main Image and Image Description At the top right of each image page is the main image, and the first bit of text on the page is the image description. The image description introduces the main image briefly, generally in one or two short sentences.

The main image is the image around which the page is organized; a page does aims to explain the mathematics behind the main image in a way that is accessible to audiences of various levels of mathematical ability. The extent to which the rest of the page focuses on the main image can vary from topic to topic. For instance, a page like Barnsley Fern (http://mathforum.org/mathimages/index.php/Barnsley_Fern) starts with an image and the rest of the page aims to explain that image in particular. By contrast, the Stereographic Projection page deploys a main image that is interesting and eye-catching, but to which the majority of the page does not refer back. An image page, then, can be organized around the main image in a robust and complete sense, or the main image can simply motivate the page.

When one is writing a Math Images page, selection of the main image is very important. In analogy to the above, it is possible for a writer to select a topic before an image or an image before a topic. There may be a captivating image whose mathematical background is worth investigating, or the topic may be decided before selecting an image to represent it. Many writers have changed the main image after working on the page for some time, as is often the case if one begins working on an incomplete page that already has an image. The process of image selection is flexible and potentially ongoing.

It is good to choose an image that is grabbing. .gif files tend to be grabbing because they are animated and therefore often can convey something about a topic that cannot be conveyed through other media, like textbooks. An example is the Taylor Series page, the main image of which shows the successive Taylor-polynomial approximations of the sine function. Since the image changes over time, the image can convey the way in which Taylor series can be used to approximate functions very closely on some interval, with the approximation improving as n increases. Another advantage of .gif files is that they may convey what is going on in the mathematics of a certain topic. An example of this is the Volume of Revolution (http://mathforum.org/mathimages/index.php/Volume_of_Revolution) page, the main image of which conveys in what sense the volume is obtained by “revolution” and why integrating over certain variables yields the desired volume.

Other image pages might use an actual photograph (Stereographic Projection). The advantage of such an image might be that the concept is made more concrete. In the case of stereographic projection, it is not entirely clear from a computer-generated image that the map can be understood as a process of drawing a line from the “north pole” of a sphere through the plane, but the concreteness of the photograph may better convey this geometrical idea.

Many pre-existing but uncompleted image pages already have main images. The writer can therefore decide whether to keep the image or find a new one. While it is possible to create an image for a page (the animated main image of the Taylor Series page was created by an undergraduate participant), there are a lot of compelling mathematical images on the internet. One should investigate an image’s terms of use before using it; if this cannot be immediately discerned, then it is often possible to email of the creator of the image and ask for permission to use it on the Math Images wiki, giving credit as appropriate.

Basic Description The first major section on the page is the Basic Description. The Basic Description is exactly what its name says—a basic description of the topic and image that the page addresses.

Most Basic Descriptions will use the main image as a springboard to discuss the topic of the page. The Basic Description may establish a broader problem to be addressed on the page. See, for example, the Systems of Linear Differential Equations (http://mathforum.org/mathimages/index.php/Systems_of_Linear_Differential_Equations) page, which introduces the Cold War conflict between the United States and Soviet Union to give an example—which is extended into the later sections of the page—of equations including derivatives (or rates) that depend on each other.

The goal of the Basic Description is to introduce the topic without rendering it daunting or inaccessible because of too much mathematical formalism. Consider, for example, the Tesseract (http://mathforum.org/mathimages/index.php/Tesseract) page. The Basic Description considers a number of ways in which someone might “visualize” a four-dimensional hypercube. This is a difficult task, but it is one of the main points of interest of the page that explains, ultimately, what the odd but interesting main image is about. The Basic Description includes a number of images and offers analogies with the three-dimensional case to facilitate understanding. There are some fairly lengthy descriptions of what is going on, but they are non-technical, and do not include a lot of equations or numbers.

A More Mathematical Explanation A More Mathematical Explanation is a hidden section that goes into more detail than does the Basic Description, often picking up where the Basic Description left off. A page’s More Mathematical Explanation may contain equations, calculations, and proofs that would complicate the Basic Description.

The idea behind this structuring of the page is that, if the page proceeds by increasing in difficulty sequentially, it will accommodate readers of various levels of mathematical ability and interest. The main image, it is hoped, will grab readers who find it appealing, and the Basic Description can help those who do not want a technical explanation to understand the topic. Most topics permit a great mathematical depth, however, so those interested can continue on into A More Mathematical Explanation to learn more.

A More Mathematical Explanation is hidden for this purpose. One should get the sense that in reading A More Mathematical Explanation, they are entering “more deeply” into the page. Note that in addition to the fact that the More Mathematical Explanation is hidden for every page, the writer of a page can choose to hide parts of the page voluntarily (http://mathforum.org/mathimages/index.php/Top_5_things_you_need_to_know_how_to_do_on_the_wiki). This can be useful if there is some especially difficult content, or if some part of the Why It’s Interesting section (to be addressed below) is in need of hiding but, according to the organization of the page, does not belong in A More Mathematical Explanation.

One page that implements some of the above is the Taylor Series (http://mathforum.org/mathimages/index.php/Taylor_Series) page. Inside A More Mathematical Explanation, there is another section Error Bound of a Taylor Series. Since this is one of the more difficult aspects of Taylor series, this subsection is hidden—despite already being in the hidden More Mathematical Explanation. However, even inside this hidden subsection, there is another hidden proof, the proof for deriving the general formula for the error bound of a Taylor polynomial. This proof is more technical than anything else on the page, and even someone who needs to know about bounding the error of Taylor polynomials may not need to know about how the formula is derived.

When hiding sections of a page, it is worth asking whether a reader could skip the hidden section an continue to read the page. A hidden section should provide an opportunity for readers with more mathematical skill to take on more challenging content, while still allowing less comfortable readers to gain mathematical insight from reading the page.

Why It’s Interesting This is the last major section of an image page. The Why It’s Interesting section offers some examples, applications, and motivations that show why the topic addressed on the page is worth being studied in itself. Here, one should try to strike a balance, since one might want to include a “hook” in the Basic Description, and one does not want to put all of the interesting content at the very end of the page.

The Taylor Series page provides an example of hidden sections in the Why It’s Interesting section. Consider the Approximating π section, for example. The unhidden portion provides some interesting historical background and includes an image that shows one method that has been used to evaluate π. It suggests that the ancient method using regular polygons, however, is rather slow and unwieldy, and that Taylor series offer a faster alternative. The hidden portion delves into this other method and how it was developed by different mathematicians over time; because the work with Taylor series and trigonometric functions is akin to the difficulty of the More Mathematical Explanation, it is hidden for those interested to pursue it.

The Why It’s Interesting sections of some other pages are not as technical. For example, Towers of Hanoi (http://mathforum.org/mathimages/index.php/Towers_of_Hanoi) has a very brief Why It’s Interesting section that notes a couple “uses” of the Towers of Hanoi puzzle and points out where the puzzle has appeared in popular culture. This is appropriate in this case because the content of the page is already fairly specific and interesting.

Why It’s Interesting can refer to a variety of areas. It may contain scientific applications of a topic or interesting but impractical corollaries. The Monty Hall Problem (http://mathforum.org/mathimages/index.php/The_Monty_Hall_Problem) is an example of a page that contains several interesting sections in Why It’s Interesting; it suggests that the controversy over the problem may have important bearing on the interpretation of results in other fields of research and probes the consequences of our limited intuitions of probability. It also manages to fit in a reference to popular culture in the movie 21.

Length Images pages vary in length. Each of the major sections should contain some content; how much depends on the nature of the page. The Basic Description should be short enough to introduce the topic in a non-technical manner. Its length can be anywhere from a single paragraph, as on the Volume of Revolution (http://mathforum.org/mathimages/index.php/Volume_of_Revolution) page to a couple pages, as on the Tesseract (http://mathforum.org/mathimages/index.php/Tesseract). Note that on the Tesseract page, the Basic Description is even broken down into subsections.

In general A More Mathematical Section will be the longest section; as a page becomes longer and continues to plumb the depths of the topic, A More Mathematical Section will probably be where most of the new content is. It can still vary a lot; for example, the More Mathematical Explanation of the Tesseract page is about as long as its Basic Description, while the More Mathematical Section of the Stereographic Projection page is quite long.

Likewise, Why It’s Interesting can vary in length and content; it might briefly survey a number of small applications and curiosities. The Stereographic Projection page focuses mostly on one topic, cartography, in the Why It’s Interesting section, but does so at length, including comparisons of stereographic projections with other maps from a sphere to a plane. By contrast, the Towers of Hanoi page, mentioned above, has a short Why It’s Interesting section.