Appearance versus Reality

Thing Appearance Reality
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Constellation Classification

I decided to classify the eighty-eight constellations into five brightness classes and five richness classes—brightness indicating the general brightness of stars in the constellation and richness indicating the abundance of interesting astronomical objects, like nebulae, clusters, and galaxies. In the brightness classes, “I” indicates a prominently bright constellation that could be easily discerned even in some urban skies,  and “V” is a “Huh, that’s a constellation?” In the richness classes, “I” indicates astronomical figures abounding everywhere whereas “V” is pretty much devoid of them. Also, color indicates where the constellation is, red being Southern, blue being Northern, and green being Equatorial. They are sorted within each cell from being-closest-to-the-next-cell-up-or-left to being-closest-to-the-next-cell-down-or-right.

In terms of star brightness, Orion is usually considered the brightest constellation of the night sky, with two (!) first magnitude stars and five second magnitude stars. Mensa exhibits the amazing achievement of having no stars in even the brighter half of the fifth magnitude or above. Sagittarius is probably the winner when it comes to astronomical objects, simply dotted with them in its northwestern portion, whereas Octans, for example, has nothing of this sort to show. (At least it can claim being the South Pole constellation.)

Proving Jo Chen’s Conjecture

A long-coveted result in mathematics is the solution to the tooth integral of the existential quantifier dotted with d7, or


After seconds and seconds of speculating upon what the correct result of this integral is, Jo Chen conjectured that


It turns out that Jo Chen correctly predicted the answer to the integral, as the solution has been found. Our first crucial step is to realize it is very difficult to conceptualize the dot product of an existential quantifier and a d7. We thus divide both sides of the equation by ·.

Canceling out the ·s in the numerator and denominator on the left hand side of the equation, we arrive at


Now comes a clever manipulation: we take the antitooth integral of both sides of the equation. It is thus now inverted at the end of the second side:


Intuition now tells us that too many things are upside down, motivation to reciprocate:


It may now be tempting to carry out the cross product, but we must be patient and deal with the fact that the order of operations requires us to deal with the leftmost operations first, as all operations involved in this expression are equally dignified. We can initially simplify by moving a denominator of the denominator to the numerator:


Then, we take care of the smiley at the start of the denominator. It is wildly obscure enough of a smiley to be well approximated by the random number function:

We now see why we don’t carry out the cross product in the first place: the expression will end up being randomized anyway. Yet our result could still be further simplified. We magnify the numerator. We can do this without magnifying the denominator because the random number function can account for all magnifications and contractions:


Notice we now have two divisions going on. Thus, we move the denominator of the denominator to the numerator. Of course, no limits of integration were provided, thus making our original integral an indefinite integral, requiring a +C. Our end result, ,

is read “A random person drowns in the C.” Since all our steps are perfectly reversible and our conjecture is equivalent to a coherent statement, Jo Chen’s conjecture has been proven to be true. //

Shakespeare Plays by Weighted Death Percentage

So while we were discussing the deadliness of Shakespeare plays, Mo mentioned that Othello actually has the highest death count because the entire Turkish fleet sinks. That brings up quite a problem, because those Turks are hardly in the plot, and thus it seems inflationary toward a death count to count them as such. Clearly, deaths should be weighted by character significance. I decided that characters’ line counts would be a good measure of how significant a character is (although clearly not perfect, it is accessible and minimally arbitrary), as if they say more, they would generally be more important to the plot of the play. It turns out Othello is still the deadliest Shakespeare play, as long as one includes Iago’s imminent death after the end of the play.

Here’s the top five plays by weighted death percentage. “Killed” in way-of-death explanation means the method of death is ambiguous.

5. King Lear—63.1% Weighted Death Percentage

Died: Lear (dies from grief)—22.7%, Gloucester (eyes gouged out by Cornwall and deserted, later “dies of shock and joy” upon finding Edgar)—10.1%, Edmund (stabbed by Edgar)—9.3%, Goneril (suicide by stabbing)—5.9%, Regan (poisoned by Goneril)—5.6%, Cordelia (hanged by Edmund’s order)—3.5%, Cornwall (stabbed by servant)—3.3%, Oswald (stabbed by Edgar)—2.4%, Other Died—0.3% (one stabbed by Regan, one killed by Lear); Survived: Edgar—12.0%, Kent—10.7%, Fool—6.5%, Albany (Goneril plotted to kill him, but never carried it out)—4.9%, Other Survived—2.8%

4. Antony and Cleopatra—63.2% Weighted Death Percentage

Died: Mark Antony (suicide by stabbing)—25.6%, Cleopatra (suicide by poison)—20.7%, Domitius Enobarbus (dies of a broken heart)—10.7%, Charmian (suicide by poison)—3.2%, Eros (suicide by stabbing)—1.4%, Iras—0.8% (suicide by poison); Survived: Octavius Caesar—12.7%, Sextus Pompey—4.3%, Aemilius Lepidus—2.0%, Menas—1.9%, Agrippa—1.8%, Dolabella—1.5%, Scarus—1.2%, Mecaenas—1.1%, Octavia—1.1%, Other Survived—10.4%

3. Titus Andronicus (how the hell did this not make first?)—78.5% Weighted Death Percentage

So yeah, if you like to be disturbed, I’ll leave you to read this play and have fun finding out how these guys died.

Died: Titus—28.7%, Aaron—14.1%, Tamora—10.3%, Saturninus—8.4%, Demetrius—3.7%, Bassianus—2.5%, Lavinia—2.4%, Chiron—2.1%, Martius—1.3%, Quintus—1.1%, Other Died—3.4%; SurvivedMarcus—10.5%, Lucius—7.5%, Young Lucius—1.8%, Other Survived—1.4%

2. Hamlet—79.9% Weighted Death Percentage

Died: Hamlet (stabbed with poisoned rapier by Laertes)—37.7%, Claudius (poisoned two ways by Hamlet)—13.7%, Polonius (stabbed by Hamlet)—8.9%, Laertes (stabbed with poisoned rapier by Hamlet)—5.3%, Ophelia (suicide by drowning)—4.3%, Gertrude (accidentally poisoned by Claudius when he meant to poison Hamlet)—3.9%, Elder Hamlet (implied to have been poisoned by Claudius prior to start of play, exists within play as a ghost)—2.4%, Rosencrantz (killed by pirates)—2.4%, Guildenstern (killed by pirates)—1.4%; SurvivedHoratio (was going to commit suicide, but Hamlet tells him someone must live to tell what happened)—7.4%, First Player—2.4%, First Gravedigger—2.3%, Marcellus—1.8%, Osric—1.3%, Bernardo—1.0%, Other Survived—3.9%

1. Othello—82.5% Weighted Death Percentage

Died: Iago (implied to be tortured and killed after end of play)—31.4%, Othello (suicide by stabbing)—25.4%, Desdemona (strangled by Othello)—11.5%, Emilia (stabbed by Iago)—7.0%, Brabantio (dies from grief)—4.0%, Roderigo (stabbed by Iago)—3.2%; Survived: Cassio—8.0%, Lodovico—2.2%, Duke of Venice—2.1%, Montano—1.8%, Bianca—1.0%, Other Survived—2.7%

Some might argue that one should only include deaths specifically within the play and not implied in the vicinity of the chronology of the play, which would make Hamlet first at 77.5%, Titus Andronicus second at 64.4%, Antony and Cleopatra third, King Lear fourth, and Othello seventh at 51.1%, dropping below Julius Caesar (fifth) and Macbeth (sixth), and only barely exceeding Romeo and Juliet (eighth).

Proof by Convergent Handwriting

Write the first line of your proof a units high. Write the next line of your proof a/2 units high. Then, write the next line a/4 units high. Continue the pattern, writing the nth line a/2^{n-1} units high. What you’re doing is mathematical reasoning, and thus you’re being rational. The set of rationals is only countably infinite, so within the infinite number of statements you make, the proof of whatever you’re trying to prove is bound to arise somewhere.

Skylular Inversion

Scroll down for the solution to the astronomy puzzle.








It becomes quite evident that Xela does not live on Earth. Rather, invert on the Earth’s sky. Then stars in the night sky map to cities on Earth and vice versa. Mapping a brighter star to a larger city, and noting whether population growth seems to be waxing/blueshifting or waning/redshifting, we figure that we should find a location in our night sky where three very bright blue stars occur in a row. Xela lives in Alnilam, Orion.








An Astronomy Puzzle

Xela frequently enjoyed wandering off into the countryside, away from the hustle and bustle of the vicinity of his urban residence. He remembered when his area was still tranquil, but recently the unbelievable population boom in his country, now one of the most populous in the world, made nature seem further and further away from life. The city he lived in was one such booming city, but the two similar cities to its west and east were exploding in population even faster. Up on the high mountains two countries to the east in the same line as that triplet of cities was yet another great metropolis, but not growing as fast. In his area of the world, there were cities everywhere. Because he loved gazing at the stars above, the deluge of lights that abounded the city saddened him greatly. Fortunately, these trips to the countryside sufficed to fulfill his desires for making friends with the stars.

He saw an area strangely devoid of stars that contrasted sharply with the splurge of bright stars to its north. In this group included many of the brightest stars of the night sky, one of which was a brilliant blue star almost in an optical double. After gazing at this wonderful assortment of lights, he turned his attention to the northeast and viewed another bright constellation, one mostly dominated by blue-white and white stars with blue stars mainly occupying a fainter magnitude. Looking further to the northeast, he finally found the brightest star of the night sky.

Even where Xela lived, it gets pretty cold at night, and yet the burning of the stars light-years away felt like it was at the same time burning a most familiar fire inside his heart that gave him a contented warmth.

Now for the puzzle: where does Alex live? Give the city and the country.

Hint: Unscramble AAIILLMNNOOR.