An Elegant New(ish) Proof
Tom Apostol’s square root of 2 proof is beautiful. This proof was published in 2000 by Tom Apostol. It just might become your new favourite proof that the square root of 2 is irrational. Here’s how it goes.
Read full storyFTL Travel and the Rear of the Moon
For our first demonstration you will need your cat, a laser pointer and the Moon. Place your cat on the Moon. (Take necessary precautions to ensure the well-being of your cat.) Fire up the laser pointer and point it towards the Moon. (You will need a strong battery or a stronger imagination. Both, if you have them.)
Read full storyAn Easter message: God is not mad at you (Opinion).
Good Friday has more to say about the power of God than the wrath of God. The Altar of the Crucifixion located in the Church of the Holy Sepulchre in Jerusalem. Foreground: The Rock of Calvary, visible through protI, Ondřej Žváček / CC BY-SA.
Read full storyThe Square Root of 2 is Irrational
Fig. 0: We can construct √2 with compass and straight-edge, Euclid-styleImage by Author. Imagine a rectangle with aspect ratio of √2. Dissect it into a grid of tiny squares: u squares long by v squares wide (Figure 1).
Read full storyA Number Theory Puzzle
(This post includes an affiliate link. If you purchase anything through this affiliate link, the author may earn a commission.) You have a numbered string of 1,000 light bulbs. You begin by flicking them each on.
Read full storyA Short Geometric Intuition of the Double Angle Identity
Amaze your friends at parties with this trigonometric proof. Freaked out by trig identies? I fee your painPixabay. Here’s the double angle identity which you may have memorized:
Read full storyA Perspective on Perspective
Become aware of your personal biases. Because — yes, you have them. “Gall–Peters projection SW” by Strebe — Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.
Read full storyA Math Puzzle For You
Find the last two digits in this expression. It’s a big number. Desmos gives up on the 3rd seven. Desmos has already copped outScreenshot from Desmos.com. Instead, let’s look at bite-sized powers of 7. (“Bite-sized” here means “less than 2 billion.”) Below are the first 12, conveniently arranged in columns of 4.
Read full storyThe Twin Paradox Explained
How special relativity predicts time dilation, length contraction and simultaneity loss. Rocket image by OpenClipart-Vectors from Pixabay, modified by Author. [Two ships in the night. Bob sees Alice travelling to the right at half the speed of light. From Bob’s frame of reference, he is at rest and Alice’s clock is running slow. Alice’s frame of reference has Alice at rest while Bob travels to the left. According to Alice, Bob’s clock is running slow. When they meet again, who will have aged more?]
Read full storyThe Law of Sines Made Clear
Once you see a triangle this way, you can never unsee it. Do you remember the Law of Sines from high school trigonometry? You may have memorized it, but did you really get it? Did you feel it in your liver?
Read full storyA Tale of Rejection
It was the summer of 1971. I was 7 years old (almost 8, thank-you-very-much). I had just read E.B. White’s Trumpet of the Swan. Finishing a novel always brought a time of grief for me. The characters who had been alive for me during those chapters were now no more.
Read full storyThe Circle-Parabola Problem
No Calculus. No Algebra. Just Plane Geometry. Actual image of Euclid solving a problem he found on InstagramWikipedia Commons. In June 2021, Dr. Peyam (of YouTube fame) published a video of a geometry problem he found on Instagram. The solution, he says, “uses a lot of beautiful geometry and even more beautiful calculus.” (It does indeed. Check it out.)
Read full storyIn the Celestial Chorus, Earth Sings Alto
From Pythagoras to Kepler, mathematicians saw music in the motions of the planets. The Earth and Venus are altos. Mercury sings soprano and Mars, tenor. Jupiter and Saturn together provide the bass.
Read full storyFind The Fox Puzzle 🦊
You have 5 holes and 1 fox. The holes are in a line, left to right, A, B, C, D and E. The fox is in one of these holes. Fox image by Daniel Roberts / Pixabay (modified by Author)
Read full storyThe Librarian
(Fiction: Unaccompanied Minor Serial, Part IV) David flinched when he heard a shrill voice fill the sacred space of silence. This was the Bent Fork Public Library. Once you stepped inside, you felt you mustn’t utter a sound beyond a voiceless whisper.
Read full storyNever Give In
“You resign now.”The Queen’s Gambit / Netflix (Fair Use) We could take the advice of Mr. Shaibel, Beth Harmon’s mentor in Netflix’s The Queen’s Gambit. Then, hanging your Queen is a good time to resign. (I can think of two sisters who would probably disagree. What would Mr. Shaibel think of the Botez Gambit? But, I digress.)
Read full storyThat Pesky Blue-Eyed Islander Puzzle: Why the solution works when it feels like it shouldn’t
"Tell us something we don't already know!"Illustration by Author. You may already know the famous (infamous?) Blue-Eyed Islander Puzzle. Here’s my version. 99 perfect logicians are stranded on a remote island. As perfect logicians, each is able to immediately determine the logical outcome of any statement.
Read full storyA Math Puzzle Involving Infinitely-Sided Dice
Here’s the challenge. You have three infinitely-sided dice. When you roll one of these dice, you get a Real Number between 0 and 1. So when you roll three of these dice, you get a Real Number between 0 and 3. Capisce?
Read full storyDandelin’s Delightful Spheres
Dandelin Spheres.Dandelion image by Capri23auto from Pixabay modified by Author. Mathematicians have been slicing open cones since the 4th Century BC. Here, we study one of these famous cross sections — the ellipse. Along the way, we’ll discover a beautiful proof from Germinal Dandelin of the 19th Century.
Read full storyMy CPAP Machine Was Recalled
The Author with his $2400 CAD paperweightAdam Hrankowski. Have you seen The Good Place? Hypatia of Alexandria points to the 5 on her shirt. “Is this an S? Or a math?” Centuries of unencumbered bliss in Heaven have rendered her and her companions intellectual zombies.
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