secrets of the universe
I tutor a nine-year-old named William who is starting university math classes in the fall. Today he thought of a connection between two concepts he’s learned recently—the difference of squares and imaginary numbers. I asked him to write down his thought process because it’s so fascinatingly advanced. The instructors didn’t aid or prompt this brainstorm in any way—he’s just that awesome.
My favorite Vi Hart video to date.
The familiar trigonometric functions can be geometrically derived from a circle.
But what if, instead of the circle, we used a regular polygon?
In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.
We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.
Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.
More on this subject and derivations of the functions can be found in this other post
Now you can also listen to what these waves sound like.
This technique is general for any polar curve. Here’s a heart’s sine function, for instance
In "California Soul (Lincoln Lawyer Remix)", Ya Boy says, “I won’t stop till my dough stretch long as a bridge.” A dollar bill is 6.14 inches and the longest bridge in California is 36,960 ft, so his dough would only amount to $72,234.53. He needs to either define additional variables or step up his hustlin’ game.
So, does this WKU staff member win the “Most Science-y Name Ever” award or what?!
What do you think is the correct way to draw a 7-way Venn diagram?
Maybe like this?
Or perhaps like this?
NOPE. Like this:
If you ever need to draw one of these diagrams, just remember that each set is in the shape of a man who is horrified by the idea of drawing one of these diagrams:
Donald Miller, A Million Miles in a Thousand Years (via amethyste-heart)
axon-axoff: I just want to say that the title of this book is really slow… (1,000,000 miles / 1,000 years) * (1 year / 365 days) * (1 day / 24 hours) = 0.114 miles/hour. That is slower that the top speed of a sloth. Is this book about a slower-than-average sloth or what.
my logic textbook is so cute