Illusions

Picture6
The Monty Hall problem. Just think about the probability that you originally picked a goat vs. the car. If you picked a goat (2/3 chance) you want to switch. So…you always switch.

Illusion, Michael. –G.O.B.

(i) Optical illusions are fun. We all have our favorites. There’s one in particular which does not get as much attention as it should. It’s called the McCollough effect and you induce it by alternately staring for five seconds or so at the image below on the left followed by the image below on the right. Repeat this procedure for about a minute or so and then look at the black and white image below the pair.

 

MC1

MC2

 

The fact that the effect can last months following sufficient induction is what makes this one so cool. And a bit scary. Sort of like hiccups that won’t go away. In the back of your mind there is a creeping fear that they may never stop. You might hiccup forty times a minute for several decades or just have the hiccups ruin your life due to an unknown brain stem tumor.

(ii) Infinity is a like a concept, not a number, brah. Yup. ∞ It’s a concept that is notoriously difficult for finite begins such as ourselves to wrap our minds around. To make matters worse (as you are probably aware if you’ve dabbled in the mathematical arts) there are infinities of different size. Now I won’t pretend to have studied or to understand set theory, but it at least makes sense in an abstract way that it would be possible to count forever by whole numbers while also counting forever by half numbers (starting with .5) and there should be more numbers counted in the second case. This next one takes it a step further.

Picture1
Cantor’s proof redrawn.

Mathematician Georg Cantor made it his life’s work to outrage his contemporaries and to blow the minds of the rest of us. Again, I won’t pretend to understand most of it, especially in formal terms, but as for one of his more visual proofs regarding infinity, I think I get it.

The basic idea is that there are an infinite amount of points on both a line of infinite extent and a finite line segment. I suppose the numerical comparison would be something like all of the natural numbers versus the all of the real numbers between 1 and 2. Did I mention that I am not a mathematician?

The proof is pretty genius. Cantor draws a perpendicular line segment down from a point on the original segment which intersects a circle between the segment and the line. What it boils down to is the fact that the angle between the diameter of the circle (parallel to the segment) and the segment drawn from the perimeter through the center of the circle, can become infinitesimally small, but as long as there is a positive value, the segment traveling through the center of the circle to the line will intersect the line at some point. So, they both contain an infinite number of points along their length. QED?

I live on the frozen surface of a fireball. –Julian Casablancas

Or, a wet rock.

The total amount of water on our planet. [https://water.usgs.gov/edu/earthhowmuch.html]
(iii) If there’s one thing that irrationally bothers me, it’s finding out that a certain factoid I’ve been proudly touting is completely false. In fact, one of the two definitions of factoid, an invented fact believed to be true because it appears in print, seems like a strange indictment of everything I am doing here.

I have dreams of writing a book called Actually… which would debunk factoids. Some examples:

“Hey did you know that window glass in old buildings is thicker on the bottom because the glass flows downward very, very slowly.” Actually… it doesn’t.

“January is named for Janus, cuz he looks back to the old year and forward to the new.” Actually… it’s named after Juno. (That source looks dicey. This one might be a double-actually.)

Proxima Centauri is the closest star to us.” Actually… it’s the sun and that’s a trick question.

Bumblebees shouldn’t actually, be able to fly.” Actually…wait, what?

I swear, I’ve heard this last one too many times and the gist of the confusion is applying (tenuous) knowledge of aerodynamics at the scale of humans and small aircraft to the much smaller world of the bumblebee. The implications of orders of magnitude differences in mass and volume on the fluid mechanics involved in terrestrial flight are profound. Particularly salient for flight are the differences in surface area. Let’s do some calculations:

A standard human length of 1.8 m versus maximal bumble bee length of .04 m = 45 times as long.

We can idealize the human and the bee as right cylinders in order to obtain rough estimates of surface area and volume.

Picture1
Human and human-sized right cylinder. Bumblebee-sized human and bumblebee-sized-human-sized right cylinder. NOT to scale.

Human: h = 1.8 m r = .25 m; volume = .35 m³; surface area = 3.22 m²

SA:V = 9.2

This Bumblebee: h = .04 m r = .0056 m; volume = .00000394 m³; surface area = .0016 m²

SA:V = 406.1

On a similar note: ever wonder why insects never get hurt when you throw them to the ground or how squirrels are so successful at avoiding death when falling out of trees? Surface area matters as it produces drag, but mass and the effect of atmospheric pressure on terminal velocity also come into play.

Let’s assume the bee and the human are equally dense, that is, made of basically the same stuff.

Human: Mass = 77 kg; [Density (Mass:Volume) = 220 kg/m³]

Bumblebee: Mass = 220*.00000394 = .00087 kg

The drag coefficient of a standing human shaped object is around 1.0.

Here’s a terminal velocity calculator to play around with. I got a human value of approximately 70 m/s whereas the actual value is closer to 53 m/s. The bee is estimate is around 11 m/s.

To sum up, from the perspective of a bee, air is much more substantial than it is from our own perspective. The experience is much more like navigating what we perceive as a dense fluid, such as water in a pool. The terminal velocity comparisons are made clearer through this analogy when imaging the action of gravity (sinking) on objects of distinct shape and mass when thrown into deep water.

[Below I uploaded a cool (related) article on the various implications of size in biology:]

Schmidt-Nielsen-1975-Journal_of_Experimental_Zoology

(iv) But back to the point. I have been going around telling everyone that the surface of the Earth is smoother than a billiard ball. I heard it somewhere and I loved it. “Guess what guys, the Earth is really, really big.” (How big!?) “It’s so big that you wouldn’t even feel the topography if you had really gigantic hands!”

I believed this hours ago. I wanted to demonstrate that it was true. I got as far as the actual smoothness of a billiard ball before I hit a dead end.

Height of Mount Everest: 8,848 m

Depth of Mariana Trench: 10,994 m

Circumference of the Earth: 40,075,000 m

Smoothness (maximum relief/circumference) = 10,994 / 40,075,000 = 0.000274

Circumference of a billiard ball: .1797 m  (diameter: 0.0572 m * π)

How smooth is a billiard ball?

This is where the trail ran cold. After some further searching I found out that someone had already addressed this. They pointed out clearly and intuitively how dumb the idea of a smoothly polished Earth actually is when you really think about it. Oh well.

(v) This one is called Courage.

 

 

 

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