April 2013

Professor Wildi’s Visualization of Units pages for unit conversions

David Cash, Mohawk College (retired), Hamilton ON

Introduction

I want to recommend an excellent resource for unit relationships and unit conversions for teaching and personal use — a downloadable set of PDF files posted on the website of the late professor Theodore Wildi of Laval University.¹

Theodore Wildi was a professor of electrical engineering who created what he called “Visualization of Units” pages for every possible unit type: e.g., mass, volume, energy, pressure, etc. In all, there are 56 of these pages, organized under eight headings, and there are also three pages explaining how to use the resource. This resource was originally published as part of a small hardcover monograph,² which I purchased in 1977. I have found it an invaluable resource ever since and have made so much use of it recently that it occurred to me to make this recommendation.

I found that Professor Wildi had updated the pages in 1988 and had posted them all on the web for download. You can locate and download the unit page(s) you want by navigating from the Wildi home page to the “List of Wildi SI Charts” page, then to the “List of Wildi SI Charts” itself and finally to each unit page you want, e.g., mass¹ (see figure, right). The pages are copyright, but there is a note on the website allowing free copying for private use and classroom distribution.³ My hard copy volume was priced at $8.50 including shipping in 1977, so I am sure this was not a commercial venture, but rather a labour of love on his part.

Page layout and conversion factors

Each unit on a page such as the mass page, e.g., gram, pound (avoirdupois), kilogram, etc. has its own rectangular box in which its full name and symbol is given (right). The units on the page are arranged from the smallest, at the bottom, to the largest, at the top of the page. SI units are always given precedence. One of the attractive features of a Wildi page is that parallel systems of units, such as the ‘everyday’ units of pounds (avoirdupois), etc., and the troy units of pounds (apothecaries’ weights), etc., may be shown in parallel columns on the page, where it is easy to see how they compare and relate.

Some of the pairs of unit boxes on a page are connected by direct vertical arrows, and each arrow has an associated numerical value. This value is the conversion factor between the two such connected units, representing the number of the lower, smaller units in one of the upper, larger unit. These numbers may be either red, being exact or by definition, or black, being empirically measured values that are not exact.

Reproduced with permission from Theodore Wildi website.

Determining conversion factors not given directly

Most pairs of units have no direct arrow connections; too many arrows would clutter the page. For example, the mass page has twenty-four unit boxes, so a total of (24 × 23) ÷ 2 = 276 unit conversions are possible. Any conversion factor not on the page can be determined from the factors that are given.

For example, suppose you wish to convert a mass in troy ounces (precious metals are valued in this unit) into grams. There is no direct arrow for a conversion between these units. You have to find a connecting pathway with arrows. Pathways may be found that are all down-legs, or may have one or more up-legs too. In this example follow all of the down-legs path:

1 ounce troy = 8 dram troy;
1 dram troy = 2.5 pennyweight;
1 pennyweight = 1.2 scruple;
1 scruple = 1.29598 gram.

Going down the page is equivalent to a multiplication. Going up is equivalent to a division.

In this example, all the factors are exact except the final value. I leave it to you to confirm that 1 ozt = 31.1035 g. The usual rules for significant figures in multiplication and division should be employed.

Uses of the Wildi visualization of units pages

The Wildi pages can be used to:

Visualize the hierarchy of all units of the same kind on a single page.
Access or determine almost any conversion factor you will ever need.
Allow a student to check a conversion by confirming that the unit with the lower numerical value is above the unit with the higher numerical value.
Learn and teach about units and the history of science and technology (e.g., newton, joule, watt, pascal, hertz, gauss, tesla, curie etc.).
Stimulate questions and discussion; provide topics for assignments and projects.

The Wildi pages cannot be used to:

Visualize the magnitude of a conversion factor, since the vertical distances are not drawn to scale.
Actually do conversions. The pages are neutral as to how you do the conversions.

Examples

We encounter a multiplicity of unit systems in science and everyday life. Being able to quickly find or determine conversion factors can make life easier and more interesting, and can even liven up our teaching. Here are a few of my favourite examples:

Having students work out the mass and volume of a million dollars worth of gold, to see if they could carry it or put it in their pocket. Gold is priced in troy ounces; the day’s price is easily found online.
Determining how many kW of electrical heat would be needed to compensate for my non-functioning gas furnace on a cold weekend. Gas appliances are rated in the power unit British Thermal Units (Btu) per hour. The Wildi power page has the required conversion factor.
Working through the conversion of fuel efficiency from miles per US gallon to L per 100 km. The Wildi length and volume (liquid) pages have the required conversion factors.

Conclusion

Conversions can be easy and fun when you know how to do them. But some of our students find conversions to be difficult. The visual approach of these pages may be helpful to students who are struggling.

Let me give professor Wildi the last words, quoting from the inside fly-leaves of the dust jacket of my printed book:

“…...If you are an engineer — you will save hours searching for conversion factors between units. …….. If you are a student — you will be happy to use the graphical conversion charts, for they make it virtually impossible to make a conversion mistake…….“