Meet The Shape That Cannot Pass Through Itself

November 17, 2025 by Donald Papp 21 Comments

Can a shape pass through itself? That is to say, if one had two identical solids, would it be possible to orient one such that a hole could be cut through it, allowing the other to pass through without breaking the first into separate pieces? It turns out that the answer is yes, at least for certain shapes. Recently, two friends, [Sergey Yurkevich] and [Jakob Steininger], found the first shape proven not to have this property.

A 3D-printed representation of a cube passing through itself [image: Wikipedia]

Back in the late 1600s, Prince Rupert of the Rhine proved it was possible to accomplish this feat with two identical cubes. One can tilt a cube just so, and the other cube can fit through a tunnel bored through it. A representation is shown here.

Later, researchers showed this was also true of more complex shapes. This ability to pass unbroken through a copy of oneself became known as Rupert’s Property. Sometimes it’s an amazingly tight fit, but it seems to always work.

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The 19th Century Quantum Mechanics

September 26, 2025 by Al Williams 18 Comments

While William Rowan Hamilton isn’t a household name like, say, Einstein or Hawking, he might have been. It turns out the Irish mathematician almost stumbled on quantum theory in the or around 1827. [Robyn Arianrhod] has the story in a post on The Conversation.

Famously, Newton worked out the rules for the motion of ordinary objects back in 1687. People like Euler and Lagrange kept improving on the ideas of what we call Newtonian physics. Hamilton produced an especially useful improvement by treating light rays and moving particles the same.

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Hackaday Links: August 24, 2025

August 24, 2025 by Dan Maloney 9 Comments

“Emergency Law Enforcement Officer Hologram program activated. Please state the nature of your criminal or civil emergency.” Taking a cue from Star Trek: Voyager, the Seoul Metropolitan Police Agency is testing a holographic police officer, with surprisingly — dare we say, suspiciously? — positive results. The virtual officer makes an appearance every two minutes in the evening hours in a public park, presumably one with a history of criminal activity. The projection is accompanied by a stern warning that the area is being monitored with cameras, and that should anything untoward transpire, meat-based officers, presumably wearing something other than the dapper but impractical full-dress uniform the hologram sports, will be dispatched to deal with the issue.

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2025 One Hertz Challenge: Estimating Pi With An Arduino Nano R4

August 9, 2025 by Lewin Day 10 Comments

Humanity pretty much has Pi figured out at this point. We’ve calculated it many times over and are confident about what it is down to many, many decimal places. However, if you fancy estimating it with some electronic assistance, you might find this project from [Roni Bandini] interesting.

[Roni] programmed an Arduino Nano R4 to estimate Pi using the Monte Carlo method. For this specific case, it involves drawing a circle inscribed inside a square. Points are then randomly scattered inside the square, and checked to see if they lie inside or outside the circle based on their position and distance of the circle’s outline from the center point of the square. By taking the ratio of the points inside the circle to the total number of points, you get an approximation of the ratio of the square and circle’s areas, which is equal to Pi/4. Thus, multiply the ratio by 4, and you’ve got your approximation of Pi.

[Roni] coded a program to run the Monte Carlo simulation on the Arduino Nano R4, taking advantage of the mathematical benefits of its onboard Floating Point Unit. It generates 100 new samples for the Monte Carlo approximation every second, improving the estimation of pi as it goes. It then displays the result on a 7-segment display, and beeps as it goes. [Roni] readily admits the project is a little too close in appearance to a classic Hollywood bomb.

We’ve seen some other neat Pi-calculating projects before, too.

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You Know Pi, But Do You Really Know E?

March 14, 2025 by Al Williams 29 Comments

Pi Day is here! We bet that you know that famous constant to a few decimal points, and you could probably explain what it really means: the ratio of a circle’s circumference to its diameter. But what about the constant e? Sure, you might know it is a transcendental number around 2.72 or so. You probably know it is the base used for natural logarithms. But what does it mean?

The poor number probably needed a better agent. After all, pi is a fun name, easy to remember, with a distinctive Greek letter and lots of pun potential. On the other hand, e is just a letter. Sometimes it is known as Euler’s number, but Leonhard Euler was so prolific that there is also Euler’s constant and a set of Euler numbers, none of which are the same thing. Sometimes, you hear it called Napier’s constant, and it is known that Jacob Bernoulli discovered the number, too. So, even the history of this number is confusing.

But back to math, the number e is the base rate of growth for any continually growing process. That didn’t help? Well, consider that many things grow or decay through growth. For example, a bacteria culture might double every 72 hours. Or a radioactive sample might decay a certain amount per century. Continue reading “You Know Pi, But Do You Really Know E?” →

How Hard Is It To Write A Calculator App?

February 16, 2025 by Al Williams 40 Comments

How hard can it be to write a simple four-function calculator program? After all, computers are good at math, and making a calculator isn’t exactly blazing a new trail, right? But [Chad Nauseam] will tell you that it is harder than you probably think. His post starts with a screenshot of the iOS calculator app with a mildly complex equation. The app’s answer is wrong. Android’s calculator does better on the same problem.

What follows is a bit of a history lesson and a bit of a math lesson combined. As you might realize, the inherent problem with computers and math isn’t that they aren’t good at it. Floating point numbers have a finite precision and this leads to problems, especially when you do operations that combine large and small numbers together.

Indeed, any floating point representation has a bigger infinity of numbers that it can’t represent than those that it can. But the same is true of a calculator. Think about how many digits you are willing to type in, and how many digits you want out. All you want is for each of them to be correct, and that’s a much smaller set of numbers.

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The Helicone: Toy Or Mathematical Oddity?

January 7, 2025 by Al Williams 7 Comments

We always enjoy videos from the [Mathologer], but we especially liked the recent video on the Helicone, a toy with a surprising connection to mathematics. The toy is cool all by itself, but the video shows how a sufficiently large heliocone models many “natural numbers” and acts, as [Mathologer] puts it, acts as “microscope to probe the nature of numbers.”

The chief number of interest is the so-called golden ratio. A virtual model of the toy allows easy experimentation and even some things that aren’t easily possible in the real world. The virtual helicone also allows you to make a crazy number of layers, which can show certain mathematical ideas that would be hard to do in a 3D print or a wooden toy.

Apparently, the helicone was [John Edmark’s] sculpture inspired by DNA spirals, so it is no surprise it closely models nature. You can 3D print a real one.

Of course, the constant π makes an appearance. Like fractals, you can dive into the math or just enjoy the pretty patterns. We won’t judge either way.

We’ve seen math sequences in clocks that remind us of [Piet Mondrian]. In fact, we’ve seen more than one of those.

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