The advent of ubiquitous networked computing provides a new medium of expression for Proofs Without Words. In this section, we demonstrate several interactive Proofs Without Words that we informally dub PWWs 2.0—the 2.0 indicating a second generation of PWWs that leverages this new medium. We discuss some of the philosophical advantages of PWWs 2.0 over traditional PWWs. At the same time, we introduce several open-source technologies that make the creation of PWWs 2.0 accessible to anyone with a computer. We finish by urging the creation of a new PWWs 2.0 column to appear regularly in this journal and can only hope that it enjoys the success of the original Proofs Without Words features in Mathematics Magazine and The College Mathematics Journal.
PWWs like Gallant's "A Truly Geometric Inequality" are powerful in their explanatory capabilities. While not changing its core approach, we argue that the PWW 2.0 in Figure 13 is even more convincing than Gallant's original PWW in Figure 12.
Figure 13. A Proof Without Words 2.0. This figure is an interactive adaptation of Charles Gallant's original Proof Without Words: A Truly Geometric Inequality. Move the black point left and right along the base to see different configurations.
This figure is rendered in a free, open-source, JavaScript-based language called JSXGraph. The mathematical symbols are generated by a free, open-source, JavaScript-based \(\LaTeX\) renderer called MathJax. Furthermore, the documentation for both of these software packages is quite good and freely available. Sketches written using these tools will run in nearly all modern web browsers on nearly all popular computing platforms with no more effort on the part of the user than simply loading the associated web page. (We note that the colors used in the figures throughout this section have been demonstrated to be friendly to readers with any of the three most common types of color blindness.)
The ability of the user to move the point left and right along the baseline generalizes the original PWW in a way that is not possible in a print medium. The interactive nature of the figure allows the user to explore this result. For example, the user may slide the point back and forth and discover that equality holds when \(a=b.\) Here we use solids and color to indicate givens, and dashes and grays to better indicate where a reader/user should focus her attention. This sort of "visual syntax" is much easier to implement in a computational medium.
The use of color also allows the reader to immediately identify the line segments corresponding to the symbols \(a\) and \(b\)—a task somewhat more difficult in the original. The reader is encouraged to work through this PWW and compare the original and the 2.0 version. For convenience, we provide a reading of this PWW tucked away in this knowl (more about knowls).
Having rehearsed Gallant's PWW in both forms, the virtue, in our view, of the 2.0 version of this PWW is that it does a reasonable job of prompting a particular set of questions, the answers to which are easily verified to yield the building blocks of a proof. By prompting questions in this ordered way, the interactive diagram has a "syntax" closer to that of a traditional wordy proof. It doesn't force the reader to go verify the labels in any particular order, but it does suggest rather specific ways to establish the labels. We think of establishing the labels as something like proving lemmas. The significance of these lemmas/labels for establishing the theorem becomes clear when we inspect the geometrical relationship between the labeled lines in the diagram. It's basically the same proof whichever lemma/label we address first, so in this PWW 2.0 we substantially address the earlier concerns that PWWs tend not to encode sufficient logical or propositional structure to count as expressing proofs in the Euclidean sense.
With the interactive nature of PWWs 2.0, sometimes we discover unexpected results. Compare the PWW 2.0 adaptation of Wolf's PWW for Viviani's Theorem in Figure 14 to the original following it in Figure 15.
Figure 14. This interactive adaptation of Wolf's PWW of Viviani's Theorem allows one to position the point \({\mathcal P}\) anywhere within the triangle \(ABC.\) (Rendered using JSXGraph and MathJax)
Figure 15. Wolf's Proof Without Words of Viviani's Theorem. [Wolf]
Because of the nature of proofs without words, we cannot know how Wolf expected the reader to get to the Viviani result, but the use of bolded, dashed and dotted lines is suggestive of a pathway. In trying to get the three perpendiculars to sum to the height of the triangle, we see that the dotted line is already in place. We are left to convince ourselves that the bolded perpendicular at \(F\) is congruent to the bolded segment \(\overline{QG}\) and that the dashed perpendicular at \(G\) is congruent to the dashed segment \(\overline{GC'}\). The three segments are now stacked parallel to the height of \(\Delta ABC\) and we are done. A closer examination of this diagram through use of the PWW 2.0 shows that this suggested pathway is not general enough.
In the PWW 2.0 version, the user is free to move the point \({\mathcal P}\). In its initial configuration (hit reload to reset), this closely resembles Wolf's original PWW. The reader is encouraged to move the point \({\mathcal P}\) in the PWW 2.0 and to observe that in general \(G\) does not lie on segment \(\overline{QC'}\) and that there is no general way to construct the dashed line segment suggested in Wolf's PWW. Of course, Viviani's theorem is still true, just not in the way suggested by casual examination of Wolf's PWW.
Additionally, this PWW 2.0 offers the user the chance to see the extremes of Viviani's theorem by moving \({\mathcal P}\) to one of the vertices of the base triangle, \(\Delta ABC\). It also lets the user ponder the meaning of Viviani's theorem when \({\mathcal P}\) is moved outside of the base triangle and to speculate on new theorems. For example, moving \({\mathcal P}\) out beyond the segment \(\overline{BC}\) might suggest that if we considered signed lengths (i.e. perpendiculars with negative lengths when \({\mathcal P}\) is on the "wrong" side of the line segment), then Viviani's Theorem still holds.
Two other free, open-source software packages called Processing and its sister project ProcessingJS provide another way of creating PWWs 2.0. Consider the 2.0 version of Kawasaki's PWW for Viviani's theorem (Figure 9) in Figure 16. Put your mouse inside the triangle and click.
Figure 16. This PWW 2.0 version of Kawasaki's PWW of Viviani's Theorem is both more general and clearer than the original. Use the sliders to complete the rotational steps, and drag the point \(P\) to reimagine the proof from a different position in the triangle. (Rendered using GeoGebra)
In this PWW 2.0, the user is able to see the animations suggested in Kawasaki's original PWW. Furthermore, the user is able to place the interior point anywhere inside the base triangle before starting the animation with a click. The sketch allows us to see the animation that Kawasaki was only able to suggest in his original PWW. Also, the freedom to place the interior point anywhere within the triangle and restart the animation makes the representation of the result more general and the "proof" more convincing. 2.0 versions of many existing PWWs could be created to reduce the concerns about the reliance on particular examples and to partially address the accusation that many PWWs attempt to encode an essentially fallacious proof by cases.
As we noted in the introduction and our brief history of PWWs, there are a number of visual proofs of the Pythagorean Theorem dating back thousands of years. There are also some excellent on-line interactive visual proofs of the Pythagorean Theorem that are of a distinctly more recent vintage.
Consider, for example, in Figure 17 the PWW 2.0 rendering of Rufus Isaacs' Proof Without Words of the Pythagorean Theorem that we showed in Figure 1 of the Introduction.
Figure 17. A PWW 2.0 adaptation of Isaacs' PWW of the Pythagorean Theorem (Figure 1). Move the endpoints of the segments \(a\) and \(b\) to change the size of the right triangle. The slider translates the pieces into a new configuration. An accompanying student worksheet is available at GeoGebraTube. (Rendered using GeoGebra)
This example is rendered in GeoGebra. GeoGebra is a well-developed, open-source, free development environment that allows the creation of rich on-line interactives. The ability to move the blue points to resize the right triangle, and then to animate the relocation of the various pieces, makes this a general and convincing proof. The use of color allows the user to keep careful track of where the pieces go. The applet allows the user to explore even the most extreme cases of this important result. It is important to note, however, that we must be skeptical that the pieces do not somehow expand or contract or change shape subtly to fit in their destination locations.
A second example, shown in Figure 18, is this applet by Jed Butler which we found on GeoGebraTube and we reproduce here for convenience:
Figure 18. An alternate PWW 2.0 of the Pythagorean Theorem by Jed Butler taken from GeoGebraTube. Move the blue endpoints of the segment and the X to change the right triangle. The slider translates the pieces into a new configuration. This proof is generally attributed to Henry Perigal and is known as "Perigal's Proof" or "Perigal's Dissection." (Rendered using GeoGebra)
The ability to move the black X, the blue points, and then to animate the relocation of the various pieces, again makes this a general and convincing proof. Though embedded here, both of these examples are actually hosted on GeoGebraTube, the free community portal dedicated to GeoGebra.
There are numerous visual proofs of the Pythagorean Theorem discussed in Roger Nelsen's books. The reader is encouraged to try to create 2.0 versions of these with some of the tools mentioned above or with one of the many other tools available.
Another exciting aspect of this new medium of expression is the possibility of rendering in 3 dimensions. While the reliability of 3D rendering on the web is still lower than we would hope at the time of this writing, it is certain to improve over time. To illustrate the possibilities, we finish with a 3D implementation of Siu's elegant "Sum of Squares" PWW which first appeared in Mathematics Magazine in 1984. For reference, we show Siu's original PWW in Figure 19 followed by the 2.0 version in Figure 20.
Figure 19. Siu's original PWW of the formula \(\sum_{k=1}^{n} k^2 = \frac{n(n+1)(n+1/2)}{3}.\) The reader is clearly prompted to animate the sequence of figures. [Siu]
Note: The following interactive component relies on a relatively new graphics engine called OpenGL. It may not work on some web browsers. If it fails to run on your computer, try to enable OpenGL. If you do not wish to do so or are unable to do so, you can still watch a video of the applet in action just below the applet window.
(i - zoom in; o - zoom out; mouse drag to orient) |
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Current Problem Size: $n=$3 [] |
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$\displaystyle\sum_{k=1}^{n} k^2$ | $\displaystyle\sum_{k=1}^{n} k^2$ | $\displaystyle\sum_{k=1}^{n} k^2$ |
$3\displaystyle\sum_{k=1}^{n} k^2 = n(n+1)(n+1/2)$ |
The implementation is done using the 3D capabilities of the Processing programming language. The user can click and drag on the animation to see all sides of the animation in progress. By allowing the user to choose a viewpoint, she can convince herself that the blocks fit together and verify that she is not viewing the diagram from a privileged perspective. In addition, this sketch demonstrates the ability of Processing to interact with other elements on the web page. For example, the user can control the problem size and restart the animation using the buttons below the animation.
It is interesting to note that Siu's PWW illustrates a common theme in PWWs that depict results in number theory—the use of blocks or squares to represent units. At the beginning of the animation, each pyramidal stack of blocks represents a sum of squares. Of course we cannot illustrate the sum of the first \(n\) squares using a pyramid. Instead, like a traditional PWW, we are forced to pick a specific value of \(n\) (in this case \(n=3\)) and let the reader generalize. However, the ability to change the problem size makes the PWW 2.0 more convincing.
This PWW 2.0 doesn't have the full generality of a mathematical induction proof, but it does have increased generality over a static PWW. One could still argue that the inspection of a finite number of figures or animations is still an attempt at proof by cases, and proof by a handful of cases is still no proof at all; it leaves just as vast an infinity of cases unexamined. The point isn't that the 2.0 version represents numerically more cases and therefore has more generality. The interesting advance over the static PWW is rather that the 2.0 version allows the user to formulate and answer questions about the generality of the representation. Which cases a user inspects will reflect her concerns about possible ways that the pattern could break down. For example, by setting the problem size to \(n=4,\) the user can address a potentially lingering concern that the result might require a different approach for odd and even numbers, as many traditional induction proofs do. By allowing the user to interact with the animation, we offer a framework for the user to challenge the claim made by the PWW and observe the results. We believe that this interactive capacity, in this case, moves the PWW 2.0 meaningfully beyond the criticism so easily leveled at print PWWs that they cannot constitute proof because proof by cases is no proof at all.
Through this sequence of examples, we have demonstrated that the barrier to creating interactive PWWs and sharing them is quite low. The multitude of free, open-source JavaScript tools grows daily. All of these tools can and should be brought to bear on the task of sharing mathematical insights. In addition, because of the openness of these tools, people interested in learning to create PWWs 2.0 can start by looking at the source code of the examples in this article—all of which are free to learn from, to use, and to modify.
Also, it is interesting to note that many PWWs enjoy rich histories. In preparation for this article, the authors reached out to some of the authors of the PWWs mentioned herein. In an email exchange with Man-Keung Siu, we learned that the origins of his Sum of Squares PWW extend back as far as the 11th century work of Shen Kuo and the later work of Yang Hui (c. 1238-1298), who, in particular, gave a formula for the number of pieces in a pyramidally shaped stack of fruit. We are grateful to Man-Keung Siu for providing this wonderful synopsis of the genesis of his PWW [Siu's email].
We are excited about this new medium of mathematical communication as both a pedagogical tool and a continuation of the historical arc of many of mathematics' most compelling visual proofs. We call for Convergence to create a regular feature called "Proofs Without Words 2.0" to receive, review and, if suitable, publish PWWs 2.0 that provide insight into and/or proof of important mathematical results. As we have seen with Rufus Isaacs' first two Proofs Without Words, there is usually some historical context associated with each Proof Without Words. Since Convergence is "Where Mathematics, History, and Teaching Interact," we feel it is appropriate for contributors to also include an historical note to accompany the PWW 2.0.
Editor's note: Convergence has a definite preference for interactive applets created using the free software GeoGebra, because these applets can be hosted by the MAA channel on GeoGebraTube. Each applet must fit in a window no wider than 680 pixels. We look forward to your submissions of PWWs 2.0!