2026-10-15 01:58:07
When I wrote about the harmonic triangle yesterday, I displayed the triangle left-aligned using fixed-width text because that was easier than producing the format you usually see for Pascal’s triangle and related triangles.
1
1/2 1/2
1/3 1/6 1/3
1/4 1/12 1/12 1/4
1/5 1/20 1/30 1/20 1/5
…
I asked Claude to write a LaTeX package to make it easier to create Pascal-like triangles and it did a great job, including handy features I didn’t think to ask for. You can download it here: numtriangle.sty.
I used this package to create the more attractive display below.
Here’s the LaTeX code that produced the image above.
\documentclass[border=4pt]{standalone}
\usepackage{numtriangle}
\NewTriangle{\Leibniz}{start=1, edges=1/\n, rule=\L - \P, fractions,
format=\displaystyle, xsep=2.8em, ysep=2.8em}
\begin{document}
\Leibniz[rows=5]
\end{document}
I then compiled the code to an SVG file with the following.
pdflatex leibniz.pdf pdftocairo -svg leibniz.pdf leibniz.svg
Bell’s triangle is not defined in a way that the LaTeX package supports, but I can produce the triangle by listing the content. I made the triangle left-aligned rather than centered just to show that this can be done.
The LaTeX code was the following:
\ListTriangle[shape=left]{
1; 1, 2; 2, 3, 5; 5, 7, 10, 15; 15, 20, 27, 37, 52}
Here’s a particularly photogenic image, Pascal’s triangle with odd numbers highlighted, showing the Sierpiński triangle pattern.
The code to produce the image:
\PascalTriangle[rows=16, mod=2, xsep=1.9em, ysep=1.3em,
cell style={font=\tiny, minimum size=1.2em},
highlight style={fill=orange!40}]
The post LaTeX package for Pascal-like triangles first appeared on John D. Cook.
2026-10-14 19:11:11
G. D. Chakerian [1] discovered the following generalization of the Pythagorean theorem.
Take any triangle and attach squares to each side. From each triangle vertex, extend a line perpendicular to the opposite side and extend the line through the square, as shown in the figure below. Then the areas of the rectangles with the same labels are equal.
It’s not clear at first why this is a generalization of the Pythagorean theorem, but if we make our triangle closer to a right triangle it’s easier to see the connection.
As the angle at B becomes closer to a right angle, the area of the orange triangles labeled II goes to zero. In the limit we’re left a theorem saying that the areas of the blue and green squares add up to the area of the square attached to the hypotenuse.
The paper [1] caught my eye because it gives a connection between a geometry problem and a circuit problem, much like this post. However, I find the theorem itself more interesting than the connection to circuits.
The first diagram is my reproduction of the first figure from the paper, with color added. I then moved the vertex A to make the connection to the Pythagorean theorem easier to see.
[1] G. D. Chakerian and M. S. Klamkin. A Generalization of the Pythagorean Theorem Seen as a Problem of Equivalent Resistances. Mathematics Magazine, Vol. 59, No. 3 (Jun., 1986), pp. 149–153.
The first line of the paper is “The following result (observed several years ago by the first author) may be viewed as a generalization of the Pythagorean Theorem.” The paper also gives two other sources of the theorem. Presumably Chakerian independently discovered a theorem that was already known.
The post A generalization of the Pythagorean theorem first appeared on John D. Cook.2026-10-14 18:00:22
What’s a connection between Edgar Allan Poe and Friedrich Wilhelm Bessel?
Poe’s story The Pit and the Pendulum describes a pendulum with a semicircular blade being lowered on a cable. This is an interesting variation on the usual harmonic oscillator because the length L of the pendulum is not constant but rather an increasing function of time L(t).
Using the small angle approximation sin(θ) ≈ θ, the equation of motion is
L θ″ + L′ θ′ + g θ = 0.
If we assume the length is increasing at a constant rate, then with a change of variables described in [1], this differential equation becomes Bessel’s equation and its solutions are Bessel functions.
The authors of [1] state that their primary interest is whether a pendulum can have the motion that Poe describes. But as they indicate in their opening sentence,
The study of the motion of a steadily descending pendulum has practical applications (e.g., to the oscillations of a load being lowered by a crane).
A more recent application would be describing the motion of a package being delivered by a Zipline drone.
[1] Robert L. Borrelli, Courtney S. Coleman, and Dana D. Hobson. Poe’s Pendulum. Mathematics Magazine, Vol. 58, No. 2 (Mar., 1985), pp. 78–83
The post Poe, Bessel, and Drones first appeared on John D. Cook.2026-10-13 21:45:18
Pascal’s triangle is the most famous triangular arrangement of numbers. I’ve written about other numerical triangles, such as Bell’s triangle and the trinomial triangle. There’s another well-known triangle that somehow I never heard of until recently: the harmonic triangle, first explored by Leibniz around 1700.
To create the harmonic triangle, start with the harmonic series
1, 1/2, 1/3, 1/4, 1/5, …
This is the first column of the triangle. Next take the differences of the terms:
(1 − 1/2), (1/2 − 1/3), (1/3 − 1/4), … = 1/2, 1/6, 1/12, …
This, shifted down one space, is the second column of the triangle. Taking differences again gives the third column, and so forth. Column c of the triangle consists of the cth differences of the harmonic series.
1
1/2 1/2
1/3 1/6 1/3
1/4 1/12 1/12 1/4
1/5 1/20 1/30 1/20 1/5
…
Each entry is the difference of the entries to its west and northwest. If we let L(r, c) be the element in the rth row and cth column, then
Every numerator is 1, though it’s not immediately obvious why this should be the case. And every number is a value of the beta function, the function that gives the normalizing constant for the beta distribution [1].
[1] Ayoub B. Ayoub. The Harmonic Triangle and the Beta Function. Mathematics Magazine, Vol. 60, No. 4 (Oct., 1987), pp. 223-225
The post Harmonic triangle first appeared on John D. Cook.2026-10-12 21:04:09
A few days ago I wrote about topological models of modal logic. If you find general topology more familiar than modal logic, this correspondence could be helpful. If you think the motivation for general topology is evident but the motivation for modal logic is suspect, the correspondence between the two fields might relieve some of your suspicion of the latter.
Topological models, however, are quite limited. You can’t model provability logic, for example, by the usual approach to constructing topological models, though there is an alternative construction.
Categorical models are less limited. You can model a wide variety of logics using categories via the Curry-Howard-Lambek correspondence. This may be cold comfort if you find category theory and logic equally mysterious. Even so, there’s something intriguing about a correspondence between two things you find mysterious.
Categorical semantics is deep and broad, but I’ll keep this post brief. The correspondence between logic [1] and categories [2] is summarized in the table below.
It’s interesting how model logic fits into this picture. Axiom K, required by normal modal logics, says
In categorial semantics, this says this says the box operator □ is a functor, mapping objects representing propositions to objects representing propositions, and mapping morphisms representing proofs to mapping morphisms representing proofs.
[1] More precisely, intuitionistic propositional calculus.
[2] More precisely, bicartesian closed categories.
The post Categorical models of logics first appeared on John D. Cook.2026-10-08 21:43:02
If you’ve followed this blog for a while, you may know that I do a lot of work with data privacy and that I have an interest in modal logic. Recently these two worlds collided: I became aware of work that uses modal logic to reason about data privacy.
The story begins with Helen Nissenbaum’s paper Privacy as Contextual Integrity [1]. Rather than simply classifying data as public or private, Nissenbaum looks at norms around the context in which data exists and moves.
… the benchmark of privacy is contextual integrity; that in any given situation, a complaint that privacy has been violated is sound in the event that one or the other types of the informational norms has been transgressed.
A couple years later Nissenbaum coauthored a paper [2] with three Stanford computer scientists using modal logic, specifically linear temporal logic (LTL), to reason about contextual integrity. The paper uses four modal operators: the usual box and diamond, plus past tense versions with a minus sign across the middle.
These four operators can be read as “henceforth”, “eventually”, “historically”, and “once.”
Henceforth means something will always be true into the future; historically means something was always true in the past.
Eventually means something will happen at some time in the future; once means something occurred at some time in the past.
You could imagine using these operators to formalize statements such as data sharing in some context is permissible (henceforth) if the data subject has (once) signed a consent form.
Understanding simple stand-alone policies does not require the machinery of formal logic, but analyzing large collections of interacting policies may. It may be, for example, that a collection of policies cannot all be satisfied, though this is not obvious from looking at the individual policies. Formalizing policies in the language modal logic makes their analysis amenable to well established algorithms.
[1] H. Nissenbaum. Privacy as contextual integrity. Washington Law Review, 79(1):119–158, 2004.
[2] A. Barth, A. Datta, J. C. Mitchell, and H. Nissenbaum, “Privacy and contextual integrity: Framework and applications,” in Proc. 2006 IEEE Symp. Security and Privacy (S&P’06), Berkeley/Oakland, CA, USA, 2006, pp. 184–198, doi: 10.1109/SP.2006.32.
The post Privacy policies and modal logic first appeared on John D. Cook.
