Yesterday I wrote a post that explored the flow of heat both forwards and backwards in time. I used this as a venue to introduce the notion of entropy and to describe one extreme example of the butterfly effect—where small changes in initial data can create big changes in the final result. That’s all fine and good and I stand by that. But I said that the reverse heat equation, which runs the flow of heat backwards in time, was an example of chaos. And as this reddit user points out, this is very wrong. I have now fixed the

# Mathematics

explanatory articles on math

##### Mathematics / Physics / Science And Math

# Heat, Chaos, and Predictability

The butterfly effect, shown comically in figure 1, is the idea that a very small change in one place on Earth can cause a very big change somewhere else. In this case, a butterfly flaps its wings and causes a tornado. This metaphor illustrates the mathematical concept of chaos, in which the Earth’s atmosphere is a chaotic system. While a single butterfly probably isn’t literally responsible for a tornado, mathematical chaos is very real and important. So this week, I’m going to try giving you some intuition for the butterfly effect using one extreme example from physics. Heat Suppose

##### Geometry / Physics / Relativity / etc.

# In-Falling Geodesics in Our Local Spacetime

My previous post was a description of the shape of spacetime around the Earth. I framed the discussion by asking what happens when I drop a ball from rest above the surface of the Earth. Spacetime is curved. And the ball takes the straightest possible path through spacetime. So what does that look like? Last time I generated a representation of the spacetime to illustrate. However, I generated some confusion by claiming that it “should be obvious” that the straightest possible path is curved towards or away from the Earth. When a textbook author says “the proof is trivial”

##### Geometry / Physics / Relativity / etc.

# Our Local Spacetime

General relativity tells us that mass (and energy) bend spacetime. And when people visualize the effect of a planet on spacetime, they usually imagine something like in figure 1, where the planet creates a “dip” in spacetime much like a “gravitational well.” But today I’m going to show you what spacetime actually looks like near a planet… and it doesn’t look anything like the common picture. This is the fifth part in my many-part series on general relativity. Here are the first four parts: Galileo almost discovered general relativity General relativity is the dynamics of distance General relativity is

##### Geometry / Mathematics / Physics / etc.

# General Relativity is the Curvature of Spacetime

Figure 1 shows light from a distant blue galaxy that is distorted into a so-called Einstein ring by the curvature of spacetime around a red galaxy. This is called gravitational lensing and today we’ll learn how it works. This is part three of my many-part series on general relativity. Last time, I told you how general relativity is the dynamics of distance, which we know is a consequence of the fact that gravity is the same as acceleration. This time, I describe the consequences of the fact gravity warps distance. And in the process, we’ll learn precisely why gravity

##### Astrophysics / Geometry / Mathematics / etc.

# Speculative Sunday: Can a Black Hole Explode?

Nothing can escape the gravitational pull of a black hole, not even light. That’s why they’re, well, black. (Of course, as I’ve described before, black holes can glow very brightly, thanks to all the in-falling matter. Sometimes they even produce gamma rays. I’m also ignoring the negligible amount of Hawking radiation that black holes theoretically produce.) Once you pass the event horizon of a black hole, you cannot ever escape. Escape is simply forbidden by the laws of physics. That is, of course…if there actually is an event horizon, not just something that looks like one. Carlo Rovelli ,

##### Electronics / Geometry / Mathematics / etc.

# Lightning Detection

Since I’ve been very busy lately my good friend Michael Schmidt agreed to do another guest post! Mike has a masters degree in physics from the University of Colorado at Boulder. You can check out Mike’s own blog at duality.io or his personal website Mike’s Personal Website. Without further ado, here’s Mike: Lightning Detection Currently, in the mid-west of the United States the first thunderstorms of the year have begun. Because I am a giant geek, I love lightning and I think tracking lightning is quite interesting. My personal favorite site is LightningMaps. On LightningMaps website you’ll see

##### Computer Related / Mathematics / numerical analysis / etc.

# Tidbit: Radio Waves Bouncing Off of an F-15

I’m afraid I don’t have time to write very much this week. So instead, I leave you with a little hint of the sort of thing I’m thinking about. The above picture is from a paper I just read. It shows a simulation of radio waves bouncing off of an F-15 fighter jet. The simulation was effected by first building the jet out of many tiny pyramids linked together at the faces (shown on the left). Then, a set of five waves or so was allowed to exist inside each pyramid. When you take all of these waves together,

##### Computer Related / Electronics / logic / etc.

# Non-Digital Computers

Non-Digital Computers This is the last installment of my many-part series on computers. Last time we used the notion of a Turing machine to define what a computer is. We discovered something surprising: that not all computers need to be digital, or even electronic! A computer can be mechanical, made of dominoes, or even just a rules system in a card game. To give you of a flavor of how inclusive the definition of a computer really is, I’ll now give you a whirlwind tour of some notable examples of non-digital computers. The Antikythera Mechanism In April of 1900,

##### Computer Related / Education / logic / etc.

# What Is A Computer, Really?

Look at the picture above. Believe it or not, that person is operating an extremely sophisticated mechanical calculator, capable of generating tables that evaluate functions called “polynomials.” Although a graphing calculator can do that, a pocket calculator certainly can’t. The device above is a mechanical purpose-built computer! This article is the next installment of my series on computing. In the previous parts, we learned about Boolean logic, the language computers think in. We then learned how to implement this logic electronically and, using our newfound understanding of electronics, how to make computer memory so that computers can record results