# Maximum and Minimum Width of Coins

Founded 27-Jul-2003
Last update 19-Mar-2005

A diameter (i.e. the maximum width) is an important characteristic of each coin. Ancient Greek coins have usually not perfect circular shape and therefore it is often useful to specify both the maximum and the minimum width.1 Maximum width of a coin is the maximum distance of two parallel straight lines that touch the coin from two opposite sides. Minimum width of a coin is the minimum distance of two parallel straight lines that touch the coin from two opposite sides.

Two examples are presented in Fig. 1 and 2 that show two coins of Kleopatra Thea and Antiochos VIII from my collection. The first coin is a silver tetradrachm struck on an oval flan (ID number KA8-AR-01). The second coin is an irregular bronze unit of the same rulers (ID number KA8-AE-01). Blue lines correspond with the maximum width and red lines correspond with the minimum width. Fig. 3 and 4 show a distance between two parallel tangents to the coins when we change their angle with respect to the first coordinate axis (the angle varies from 0º to 180º).2

Fig. 1 and 2: Parallel straight lines which define the maximum and minimum width

Fig. 3: Distance between two parallel tangents making a given angle with the first axis (AR tetradrachm)

Fig. 4: Distance between two parallel tangents making a given angle with the first axis (AE unit)

It is interesting that there exists the following simple relation between the circumference of a coin and its average width: π × average width = circumference.3 It is also interesting that if the maximum width is equal to the minimum width, then it does not necessarily mean that the coin is circular. Indeed, there exist shapes of constant width other than the circle, e.g. the so-called Reuleaux triangle.4

Note that coins are three-dimensional objects so the above definitions of the maximum and minimum width are not completely correct. Nevertheless, from the practical point of view they seem to be sufficient. Note also that, of course, there is no connection between a beauty of a coin and regularity/irregularity of its flan.

1 There is also an opinion that, in most cases, it is sufficient to specify the horizontal diameter across the center of the flan with the top of the obverse in the upright position. See Hauck, James A.: Ask the Experts. The Celator, Volume 17, No. 4 (April 2003), p. 48-49.

2 The width for different angles were not measured on the coins because such way is cumbersome and non-accurate. Only the maximum width was measured on each coin and then a photo of the coin was processed by a computer. The measured maximum width was used for proper scale calibration during the computation.

3 Formally, the equation has the following form: ∫ Width(α) dα = circumference, where the integral is from 0 to π and Width(α) is the width for angle α (i.e., the distance between two parallel tangents making angle α with the first axis). This formula is a special case of so called Cauchy-Crofton formula used in differential geometry. Of course, this equation is valid only if the planar projection of the flan is convex. If a coin has a non-convex shape (as in the case of the bronze coin on Fig. 2), then it is necessary to replace the circumference of the coin in the formula by the circumference of its convex hull.

4 See, e.g., Eric W. Weisstein, Curve of Constant Width and Reuleaux Triangle (MathWorld – A Wolfram Web Resource).