And now it’s all this

The indispensable Michael Tsai has an interesting post up today about Apple Park and whether its design isolates the people who work there, despite its having been designed—in part, at least—to encourage collaboration. As usual, Michael has collected a group of choice quotes on the topic. I won’t link to any of them. You should just go to his blog, read the excerpts he’s assembled, and follow the links to see more. But I’ve often wondered how good Apple Park is as a working environment,…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Yesterday I watched the most recent Numberphile video, in which Ben Sparks explains a few finite series and the equations that simplify the calculation of their sums. He derives the formulas graphically, and it’s all very cleverly done, especially the one for the sum of cubes. The idea is to get a simple polynomial expression in n for N=∑m=1nm3 As I said,…

A couple of days ago, Casey Liss took a break from arguing about temperature scales to tweak me about the recent passage of the Sunshine Protection Act by the House. The Act would make Daylight Saving Time permanent, something Casey knows I disapprove of. A similar bill passed the Senate a few years ago, and Donald Trump has said he will sign this one, so there’s a decent chance it’ll become law. Let’s see what will happen if it does. First, of course, there will be a lot of cheering from the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] You’ve probably seen somewhere that the density of Saturn is less than that of water. If there were a bathtub big enough to hold it, Saturn would float. I think I first read this in one of Isaac Asimov’s collections of science essays. If you do an image search, you can easily find many illustrations of Saturn floating in water. Most of these show no more…

I saw this article at Inside Higher Ed this morning, guided by a Mastodon post from Techmeme. The title of the article is “Brown Professor Suspects Majority of His Class Used AI to Cheat,” so if you’re sick to death of reading about AI—pro, con, or caveated—don’t feel obligated to follow the link. I’m interested in a plot included in the article more than the article itself. An economics professor gave his class a take-home midterm, and the grades on it were much higher than usual. He suspected…

There’s been a lot of talk lately about Mac application icons and “squircle jail.” Inspired by this post from Paul Kafasis on the Rogue Amoeba blog,1 many Mac-adjacent people have taken up his cause to “Free the Icons.” I agree, but Apple’s 50th anniversary has gotten me thinking a lot lately about the early days of the Mac, so it’s only natural that my mind shifted to the highly constrained icons Mac applications had back then. In those days, icons were 32×32 pixel images, and every pixel was…

I visited my eighth and final Louis Sullivan jewel box bank yesterday morning. The Purdue State Bank (now a Chase branch) is in West Lafayette, Indiana. As a graduate of the University of Illinois, I have thoughts about Purdue University and its substandard engineering program, but I will keep those thoughts to myself and focus on the bank. This is the north side of the bank. The thoroughly incompatible stone addition to the building was (according to Wikipedia) built in the 50s, but I must say…

Yesterday I visited the Home Building Association Bank in Newark, Ohio, a Louis Sullivan jewel box bank built in 1914. It’s currently owned by the Licking County Foundation (oh, grow up), which restored it at considerable cost over several years and reopened it to the public last fall. As you can see from the photo above, the Old Home differs from the other jewel box banks in that its exterior is clad entirely in terra cotta—it’s not mostly brick with terra cotta accents. But the accents still…

Last year, around Labor Day, I visited five of Louis Sullivan’s jewel box banks: Farmers and Merchants Union Bank in Columbus, Wisconsin National Farmers’ Bank in Owatonna, Minnesota Henry Adams Building in Algona, Iowa Merchant’s National Bank in Grinnell, Iowa People’s Savings Bank in Cedar Rapids, Iowa These all fit in with a little driving circuit that included visiting my daughter and my younger son. This year, I headed east to pick up the last three. This morning’s bank was the People’s…

This week’s episode of Upgrade (which I listened to on a longish drive yesterday) has a preview of Jason Snell and Myke Hurley’s upcoming Designed in California podcast. No, not the “Road to the Apple II” series they’ve been squeezing into the Upgrade feed these past few weeks. This sneak preview—with special guest John Siracusa—is about the state of the classic Mac OS in the late 90s. In a word, the state was sorry. This was largely due to some expedient decisions made in the early 80s that…

I’ve been thinking about 1984 lately—the year, not the novel. What got me thinking about it was the reduction in gas prices over the past few weeks, and this article in the New York Times about Donald Trump’s declining poll numbers among white working-class voters and how that might affect November’s elections.1 Forgive me for not being especially optimistic, but one of my distinct memories from 1984 makes me think the polls are lagging. A news story on one of the networks back in 1984 included…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] At the end of yesterday’s post about the combinatorics of the World Cup group stage, I said “this is good enough for me.” Turns out that was a lie. Today I rewrote some of the code to come up with a slightly more detailed result. The last bit of work in that post was to generate the 40 unique point totals that can come from a group. These are the possible…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] We’re in the middle of the first set of games in the group stage of the 2026 World Cup, and I’ve been thinking about how many ways the points can be distributed among the teams in a given group. I used Python to help with the enumeration. Here’s a quick summary of how the group stage works: The teams are split into groups of four. Within each group, all…

I took a bike ride this morning on a portion of the I&M Canal Trail from Romeoville to Joliet and back. It’s a fairly short ride, eight miles each way, and that distance got me thinking about using Mathematica to do some calculations after I got back home. The trail has mile markers with little snippets of information about the canal and the surrounding area. Here are the markers I passed this morning: The mileage figures on the markers increase as you go south and west, which is downstream.…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] After my last post—the one about using Fourier series—I started thinking about how to use Mathematica to develop Fourier series.1 I could, of course, use the Integrate function to determine the Fourier coefficients, but Mathematica has other functions that can do the job directly once you understand how they work. Mathematica has several Fourier functions,…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] We’re finally here, at the end of all things. In this post, we’ll use a Fourier series to get the formula for the center deflection of a simply supported beam with a uniformly distributed load. We’ll see some of the same math that we saw in the previous Fourier series solution, but the fundamental approach will be different. Let’s start with the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] This would have been the last post in the series if I hadn’t realized recently that another method deserved a post. So this is the penultimate. It was described in this 1943 paper by Nathan Newmark and is known—or was known—as Newmark’s method. I say was because this method is dead. How dead? Forty-five years ago, when I was an undergraduate taking…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] We’re in the home stretch of this series, so ANIAT will soon go back to complaining about Apple’s UI choices.1 Next week’s WWDC keynote should provide some inspiration. But today’s post covers our eleventh method for deriving the center deflection of a uniformly loaded simply supported beam: the dummy unit load method. When I was an undergraduate, this was…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The biggest problem I face when writing posts in this series is deciding how much background explanation to put in. If I included only the stuff I do to get the answer, the posts would be very terse: one sketch and a dozen or fewer lines of math. A full explanation, though, could easily turn into several semesters’ worth of structural analysis theory. I’m…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Continuing our series on the many ways to get the center deflection of a uniformly loaded beam, we come to another energy-based technique: Castigliano’s method. Castigliano’s second theorem provides a relationship between displacements, forces, and the strain energy in a linearly elastic system. The strain energy, U, is the potential energy of the system…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Today we’ll use the Rayleigh-Ritz method again, but this time we’ll avoid dealing with an infinite sum. In case you’ve forgotten, this is our problem: We’ll express the shape as a polynomial. The form of the governing differential equation, EIyiv=w tells us that our solution won’t have any terms of higher power than x4. That gets rid of the infinity…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] I just learned that people are listening to music pitched slightly down because it makes them feel better. Instead of the A above middle C being set at 440 Hz, they have it tuned down to 432 Hz. This strikes me as odd, but how you feel is how you feel. Do whatever you want, as long as it doesn’t hurt anyone. I was interested, though, in the math behind…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Continuing our trip through various methods to derive the equation for the center deflection of a uniformly loaded simply supported beam, today we’re going to do the first of two solutions using the Rayleigh-Ritz method. Of all the possible shapes a beam can deform into, the shape it will deform into is the one that minimizes the potential energy of the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The sixth way we’ll derive the formula for the center deflection of a uniformly loaded simply supported beam is the Myosotis method, which I wrote about over a decade ago. This is the method popularized1 by J.P. Den Hartog in his Strength of Materials textbook. Image from Wikipedia. Myosotis is the genus of the forget-me-not flower, and the idea behind the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The next technique we’ll use to derive the formula for the center deflection of a simply supported beam with a uniform load is the slope-deflection equation: MA=2EIL(2θA+θB−3ψ)−FEMA Let’s start by explaining where all the terms come from. Here’s a beam of length L with arbitrary end supports (could be simple, fixed, free, or sprung) and an arbitrary…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The fourth way we’re going to derive the formula for the center deflection of a simply supported beam with a uniform load is the conjugate beam method. This is probably tied with the moment-area method for the simplest and fastest way to get the formula—at least if you’ve memorized the properties of parabolas. I wrote about the conjugate beam method last…

In Paul Krugman’s post today, he includes two charts. One is fine, the other isn’t. Here’s the first one: Generally, I’m not a fan of putting stuff in the margins that could be within the body of the plot itself, but at least it’s clear which label goes with which data series. Too bad that’s not the case with Chart 2: The two series end at almost the same spot, so putting the labels out in the margin means they can’t both be centered on their series. The little leader lines that run from the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Continuing our odyssey through various ways of calculating the deflection at the center of a simply supported beam with a uniform load, today we use the moment-area method. There are two moment-area theorems used to calculate the slopes and deflections of a beam. Here’s how they’re given in the textbook I used as an undergrad, the 3rd edition of Elementary…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] In the introductory post to this series, I mentioned that the shear is the derivative of the moment. It’s easy to see that for our specific problem, the simply supported beam with a uniform distributed load, but it’s also true in general. A further relationship is that the distributed load function—let’s call it q—is the derivative of the shear (with a…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Here’s the first of the derivations for the center deflection of a simply supported beam with a uniform load. We start with the differential relationship between the bending moment, M, and the deflection, y: M=−EId2ydx2 The second derivative of y is the curvature of the beam (for small deflections, which is one of the fundamental assumptions of beam…