*An 1869 chart by Charles Minard (1781-1870) depicting Napoleon's 1812 Russian Campaign. The majority of articles about informational graphics contain this figure. This chart combines data on the number of men in Napoleon's men, their movements, as well as the temperature they encountered on the return path. This chart quite effectively represents the army losses.(Via Wikimedia Commons. Click for larger image.)*

I thought about information graphics when I was doing some experimental mathematics on an interesting pair of triangles. I was introduced to these triangles by a posting on Evelyn Lamb's Roots of Unity blog.[3] She writes that there is only one pair of triangles, one right triangle and one isosceles triangle, that have integer side length, equal perimeters and equal areas (see figure). Lamb gives some background on this problem, which relates to algebraic geometry.[3]

*A right triangle and an isosceles triangle with the same perimeter and area.In each case, the perimeter is 864, and the area is 23,760.(Created using Inkscape)*

I decided to examine the area of all triangles with the same perimeter intermediate between these two triangles; that is, I examined all triangles with sides a = [132..135], b = [352..366], c = [366..377], selected those with perimeter 864, and calculated their areas. There are 46 triangles that meet the perimeter criterion, and these were discovered with a simple computer program (source code here, and the resultant data file here). Finding the area is easy if you know the height (altitude) of the triangle, since the area is given by the formula

Where

This equation can be written in terms of the sides, only, as

As a first visualization, I wrote a computer program that created an SVG image of all the triangles (source code here, and a required data file of apex data is here). Drawing the triangles required a calculation of the positions of the three apices of each triangle. This was done using the Gnumeric spreadsheet application. While two of these apices are trivial, since they are at the ends of the base, one of them is at the top of the triangle altitude. A little bit of algebra using the Pythagorean theorem found that

*Triangles with a perimeter of 864 with sides in the range of a = [132..135], b = [352..366], and c = [366..377]. The special pair of triangles is shown in red. (Generated by a computer program, as explained in the text. Click for larger image.)*

So, how does the area vary over these triangles? Although the data could be presented as a table, the table would have 48 rows, and it would be difficult to discern any patterns. That's why informational graphics are important. I tried various ways of graphing the data, and I decided that the column chart, shown below, is the most informative. It's also aesthetically pleasing, and the pattern is reminiscent of the reflectors of a Yagi antenna.

*Column chart of the area data, as sorted in increasing a, then b, values. Since the differences in area are small with respect to the average area, the differences from the "ideal" area are shown. The locations of the isosceles and right triangles are shown. (Click for larger image.)*

I tried several other graph types. The radial and surface graphs, shown below, don't provide as much information. The radial graph is interesting from an artistic perspective, and it makes me wonder how much good art might be lurking in mathematical problems like this.

*Like a dove. On the left is a radial plot of the area data, looking very much like a dove. On the left is a surface plot of the area in terms of side lengths b and c.Mathematics has enriched art through such concepts as perspective, and many artists have created geometrical works. One example is Josef Albers (1888-1976), who created a series of paintings called "Homage to the Square." (Both plots rendered using Gnumeric.)*

I decided to find how close a triangle could come to the area of an equilateral triangle with sides 200 and perimeter 600. I checked all triangles with sides a = [180..220], b = [180..220], and c = [180..220]. A triangle with sides 188-211-203 comes within −0.000890% of the 17320.508076 area of the equilateral triangle. Alas, the perimeter of this triangle is 602.