Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being ~Gottfried Wilhelm Liebnitz One of the first things we learn how to do is multiply numbers. . That sort of thing. But what if we multiply a number by itself? This is the familiar operation, which we call squaring a number. and . That sort of thing. You can take a number to a power by multiplying it by itself some number of times equal to the power. So if you square a number, you’ve taken it to the second power.
Mathematics
explanatory articles on math
Discrete Math / Mathematics / Science And Math
Probability: Part 1
Hi everyone! This week, I was traveling to Park City, Utah, to participate in the 3-week Park City Mathematics Institute. It’s currently a blast! I have more time now, but in the meantime, I asked my good friend Mike Schmidt to write a guest article for me. He wrote on probability which, if you’ve been reading for a while you know, is deeply connected to modern physics. Anyway, here’s the article. Thanks, Mike! The laws of Probability So true in general So fallacious in particular. ~Edward Gibson Probability is just so fantastic, I could eat it all up.
cosmology / Geometry / Mathematics / etc.
A Mess of Cosmic Coincidences: Problems With the Big Bang Theory
The furthest bodies To which man sends his Speculation, Beyond which God is; The cosmic motes Of yawning lenses. ~Robert Frost, I Will Sing You One-0 I apologize for the long time of silence! I graduated from the University of Colorado about a month ago and was immediately assaulted by a huge amount of family affairs… and then caught up in moving. Sorry about this, everyone! My regular Sunday update schedule should resume next week. Last time, I described the theory of the Big Bang. I gave some history of the theory, and some reasons for why we believe
Geometry / Mathematics / Physics / etc.
An Update On Kaluza-Klein Theory
I recently posted an article on Kaluza-Klein theory. This was partly because I was working a paper on it as a final project in my second semester of general relativity. The paper is finished, and I thought I’d upload it for the more mathematically inclined of my readers. If you’re interested, you can find it here.
Computer Related / Discrete Math / Mathematics / etc.
R.I.P. Kenneth Appel
Imagine that you’re a stingy cartographer and that you want to make a colored map of the united states. Because you’re stingy, you want to avoid spending money on ink. You have to color the map so that no two adjacent states are the same color—otherwise you wouldn’t be able to tell them apart! If you want to buy the fewest colored pens possible, how many colors must you use to make your map? Very early on, mathematicians guessed that the answer was four colors. However, no one could prove it. An example map is in the tittle figure,
Geometry / Mathematics / Physics / etc.
Stuff From Shape — Kaluza-Klein Theory
There is geometry in the humming of the strings. There is music in the spacing of the spheres. ~Pythagoras When Albert Einstein and David Hilbert published the theory of general relativity, they weren’t just proposing a new theory of gravity. They were proposing a new way of thinking. In general relativity, gravity isn’t a force. Instead, it’s a natural consequence of the shape of the universe. Force comes from stuff. Matter pushes and pulls on other matter. A proton may need to use its electric field to attract an electron, but the field is a property of the proton.
cosmology / Geometry / Mathematics / etc.
Receding Horizons: Dark Energy and the Expanding Universe
Astronomy compels the soul to look upwards and lead us from this world to another. ~Plato The history of astronomy is a history of receding horizons. ~Edwin Powell Hubble Last week, I discussed the possible shapes our universe could take. I offhandedly mentioned that not only is the universe expanding, but that that expansion is accelerating. We attribute this expansion to a mysterious phenomenon we call dark energy. This week, I want to explore the history of this idea and the beautiful experiments that tell us all is not as it seems. The Static Universe and Einstein’s Greatest Blunder
cosmology / Geometry / Mathematics / etc.
For There We Are Captured—The Geometry of Spacetime
All about me there are angles— strange angles that have no counterparts on the earth. I am desperately afraid. ~Frank Belknap Long, The Hounds of Tindalos Whoever…proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured. ~Albrecht Durer I was recently asked: What does it mean when we say spacetime is “curved” or “flat?” The answer lies in the interface between differential geometry and physics. This is the latest in many articles I’ve written on Einstein’s relativity, so you might want to check out my series on faster-than-light
Geometry / Mathematics / Science And Math
TexTAG Conference Report
Hi everyone. Sorry, but I’ve been at the Texas Undergraduate Geometry and Topology conference all weekend and I haven’t had time to write my blog post yet. I will post actual content as soon as I can, probably late tomorrow afternoon. In the meantime, I gave a talk on differential geometry at the conference! It’s not much of a consolation prize, but here are my slides from the talk. To better see the families of commensurate curves on various surfaces, we can animate our numerical integral as a function of the initial conditions. Here are some of the animations:
Geometry / Mathematics / Physics / etc.
A Space-Time Cocktail: Minkowski Space and Special Relativity
Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. ~Hermann Minkowski Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore. ~Albert Einstein In my previous discussions of how we know the speed of light is constant and how this results in special relativity, I used Albert Einstein’s thought experiments to derive the time-dilating, length-contracting results. There’s another way to describe special relativity, though, invented by the Polish mathematician Hermann Minkowski. It