Bondi k-calculus is a method of teaching special relativity popularised by Sir Hermann Bondi, that has been used in university-level physics classes (e.g. at The University of Oxford), and in some relativity textbooks. The usefulness of the k-calculus is its simplicity. Many introductions to relativity begin with the concept of velocity and a derivation of the Lorentz transformation. Other concepts such as time dilation, length contraction, the relativity of simultaneity, the resolution of the twins paradox and the relativistic Doppler effect are then derived from the Lorentz transformation, all as functions of velocity. Bondi, in his book Relativity and Common Sense, first published in 1964 and based on articles published in The Illustrated London News in 1962, reverses the order of presentation. He begins with what he calls "a fundamental ratio" denoted by the letter (which turns out to be the radial Doppler factor). From this he explains the twins paradox, and the relativity of simultaneity, time dilation, and length contraction, all in terms of . It is not until later in the exposition that he provides a link between velocity and the fundamental ratio . The Lorentz transformation appears towards the end of the book. History The k-calculus method had previously been used by E. A. Milne in 1935. Milne used the letter to denote a constant Doppler factor, but also considered a more general case involving non-inertial motion (and therefore a varying Doppler factor). Bondi used the letter instead of and simplified the presentation (for constant only), and introduced the name "k-calculus". Bondi's k-factor Consider two inertial observers, Alice and Bob, moving directly away from each other at constant relative velocity. Alice sends a flash of blue light towards Bob once every seconds, as measured by her own clock. Because Alice and Bob are separated by a distance, there is a delay between Alice sending a flash and Bob receiving a flash. Furthermore, the separation distance is steadily increasing at a constant rate, so the delay keeps on increasing. This means that the time interval between Bob receiving the flashes, as measured by his clock, is greater than seconds, say seconds for some constant . (If Alice and Bob were, instead, moving directly towards each other, a similar argument would apply, but in that case .) Bondi describes as “a fundamental ratio”, and other authors have since called it "the Bondi k-factor" or "Bondi's k-factor". Alice's flashes are