# Astronomical, Mathematical, & Logical Oddities

For each question, just hit on the link-word(s) to find the answer.
[Most of these items are not original with DIO. Those that are or might be (several most probably aren't) are asterisked.]

• What is the nearest star to us?

• *How does one crudely measure the difficulty of a shot in a game of pool, as a function of distances and angle?

• What's got twelve inches, and a foot on both sides?

• You are told that a closed room contains 3 light-bulbs, 1, 2, & 3, connected randomly to 3 exterior On-Off switches: A, B, & C. You are asked to manipulate the switches as you like, and then enter the room and identify which switch goes with which bulb.

• While using the Big Dipper's pointer-stars is well-known, other simple methods for finding north can be found. DR's favorites can be used to find Polaris even from a moving car, using strictly 1st & 0 magnitude stars:
*In Summer, complete a rhombus from the Summer Triangle and extend ordmag 10%. *In Winter, extend the Rigel-Capella line about 3/4 its own length.
Remember that Polaris is (unlike the slightly dimmer false-polestar Kochab) alone (i.e., has no comparably bright stars near it), and is always (within a degree) L° above your horizon, where L° is your latitude.
[Keep something else in mind, which may help inspire you to learn outdoor astronomy. An experienced astronomer can find rough directions from a mere glance at the Sun or the Moon or a planet or a constellation — and can usually do so even under very spotty seeing conditions.
(The following item provides a somewhat slower method but one which can be understood even by one totally inexperienced in astronomy.)]

• *A handy starbased DIO orientation-method actually requires no knowledge of stars:
Prop your head (and thus eyes) against a tree (or other steadying object) and observe carefully the diurnal motion of an overhead star. Its line-of-motion will be east→west, so a perpendicular to it is north. (I.e., if the motion's direction is mentally turned into an arrow, take it on a 90° left turn — and the arrow will be pointed north.)
[This should allow quick (under 5 timemin) identification of Polaris, a lone prominent (2nd magnitude) star which will (for next 2 1/2 centuries) always be within one degree of the north celestial pole (NCP). The NCP has a true altitude equal to your geographical latitude and will be directly above your true north.]
If one is carrying a walking stick (or book or somesuch) and can prop it up (in low tree-branches, say), this can be used as a mobile horizontal straight-edge, which can be re-set until the star seems to stay visually clinging to that edge even while moving — which establishes that the edge is oriented E-W.
[Ordmag 1° orientation accuracy can easily be achieved in a fraction of an hour, even if Polaris is cloud-covered. In the case of severe but non-total cloudiness, any star spotted may be tracked for directional clues: if the star is rising, it's in the east half of the sky: if declining, west half; if neither, it's on the N-S meridian.]

• What celestial constellation is a torture-murder device?
(No, there's no tobacco-constellation.
[Unless you count Indus the American Indian, who invented the weed's use as a drug.]
Try again.)

• What is the name of the curve which a thrown rock's path follows?

• Add integers from one to N. Square the result to get X. Add cubes from one to N to get Y. Y will equal X.

• How did the ancient Greek genius Aristarchos of Samos use an ordinary-seeming observation such as the half-Moon to reveal man's [a] non-central place in [b] an enormous universe?
(See DIO 4.2 [1994] ‡9 §K13 [p.84].)

• On which planet can one see the Sun rise, turn about, and then shortly set near where it rose?

• Hooke's law (F = kr+1) or (also due to Robert Hooke: see Nick Kollerstrom The Dark Side of Isaac Newton 2018 pp.69&88) “Newton's law” (F = kr−2) ensure “closed” orbits, using F = force, r = distance from center of gravitation (and k just a constant). A “closed” orbit means that the orbiting body returns regularly to any spot it ever occupied — and moving with the same velocity vector it previously possessed there. If we refer to the exponent as n, then what other values of n allow closed orbits?

• What part of a (traditional pagan) constellation was swiped by Christians to form Christianity's brightest celestial symbol, Crux (“Southern Cross”)?

• Not So Factual Facts:
The 2006 Pocket Idiot's Guide to Not So Useless Facts p.158: “Since it was discovered in 1930, Pluto has moved less than one third of the way around the Sun in its estimated 248-year orbit.”
The authors' reasoning: 76y/248y = 31% < 33%.
What is the truth?

• What legendary astronomer's name can literally mean “horse's ass”?
[You may or may not want to know. But there's no may-be on one point: you don't learn this stuff anywhere else.]

• Call q the inverse-square of any given integer n. And call M the SUM of the q of all integers n from 1 to ∞. [I.e., add 1 + 1/4 + 1/9 + 1/16 + 1/25 + ….] Then take the PRODUCT of the (1 − q) for all primes from 2 to ∞; call that result D.
[I.e., find (3/4)×(8/9)×(24/25)×(48/49)×(120/121)….
*Note: D is the limit (as we go to infinity) on the probability that two large numbers Don't share a prime factor.
(DIO 11.1 [2002] ‡3 §E2 eq.4 [p.24].)]
The product of M and D will be found to be exactly unity.
[M is 6 into the square of π.  D is 6 divided by the square of π.]

• What celestial object is named for an English king?
Cor Caroli, the brightest star in Canes Venatici.

• There are two twin-planet pairs in the Solar System. The terrestrial planets Venus and Earth have nearly the same mass and volume. The giant planets Uranus and Neptune have nearly the same mass and volume. (These two giants are each an ordmag more massive, and two ordmags more voluminous, than either of the two terrestrial planets.)
*Question: What are the several strange similarities which the pairings have to each other?

• *How often does the Sun set on Venus?

• Who first realized that Neptune's orbital period is 165y?

• Bugs are placed in a bottle at 7 o'clock. Their population doubles every minute. By exactly 8 o'clock, the bugs have completely filled the bottle.
What time was it half-full?

• *How many times dimmer is sunlight on Pluto than full-moonlight on Earth?

• What does ii equal?
[Thanks to my Gilman School classmate and fellow-mathematician Armand Girard for pointing out this gem.]
For clue to solution, see below.

• *Who discovered the large Uranian satellite, Umbriel?

• How many times harder does the Earth pull on the Moon than does the Sun?

• Why is a manhole cover circular?

• *What is the largest number of feet a table can have and yet find a stable (non-rocking) setting on an uneven floor?

• Which planet does Pluto get closest to?

• Ever seen an artificial satellite?

The Sun. (Alpha Centauri's companion, Proxima Centauri, is 2nd nearest.)

For a pool-shot where a is the distance from cue-ball to object-ball, b is the distance from the latter to the intended pocket, and θ is the angle between these two lines, the shot's difficulty D is roughly:

D = ab/cos(θ)

(This ignores various other factors, such as the aspect of approach to a pocket, table-tilt, etc.)

A yardstick.

Simple: turn switches A & B to On for a minute. Then turn B to Off. Upon then promptly entering the room, you know that the lit bulb correlates to switch A; and, of the two non-lit bulbs, the warm one correlates to switch B.
(With good timing and sensitivity to heat, one could extend this method to correlate four (easily) or even five bulbs pretty reliably.)

Crux — the [Southern] Cross.

No, it's not a parabola. Try again.

Near the terminator on Mercury near perihelion.
[This was realized soon after the explosion of the durable myth that Mercury kept the same face to the Sun, and it was realized that Mercury instead synchronously rotated thrice while revolving twice. This meant that (since its orbital eccentricity is slightly greater than 1/5) its perihelion speed exceeds the mean speed by more than (5/4)-squared, a factor which itself exceeds 3/2.]
[The old error is preserved with deliberate anachronism in the 1982 Baltimore-based B.Levinson film Diner, one scene of which was filmed in DR's Roland Park neighborhood, right on Roland Avenue.]

Celestial Billboards: Rearing the Mod God Bod:
Christians' brightest asterism Crux was — aptly some might say — formed from the west end of an eastbound horse.
[OK, so Centaurus is part human. But the part which Christians grabbed to form Crux (the Southern Cross) was all horse.
Have Christians ever previously been made aware of the potential here for pagan-snickers? DIO is ever pioneering….] The bright but small constellation Crux [Southern Cross] is a post-classical-Greece invention, rearing into the sky a permanent free ad for the Christian business. (In the northern sky, Cygnus' main stars were less officially converted into the “Northern Cross”, by turning a bird-rump into Jesus' brain and a bird-bill into his nailed feet.) Crux's few stars (displayed upon the flags of Australia & New Zealand) include the closest pair of 1st magnitude stars in the sky. Crux's stars were, back in the pre-Christian era, the Centaur's hind legs.
I.e., when those puckish pagans ever-so-generously relinquished ancient stellar-turf to the modern Christians, what had borne an old-mythic horse's ass was replaced by what had reared the very-dead body of the mod-cult-mythic “god”….

Due to the huge (25%) eccentricity of Pluto's orbit, it moved unusually quickly between 1930 and 2006: its heliocentric longitude increased 156° (from 109° to 265°); and 156°/360° = 43% > 33%.

Hipparchos of Nicaea, whose best astronomy was carried out near Lindos on the island of Rhodos.
(Is the tiny post-Hipparchos constellation Equuleus a memorial to him? — whether meant seriously or [pre-Crux] jocularly?)
[“Hipp” = horse; and the Greek word “archos” (“archon” is from the same root) can mean either leader or rectum. (Thus, the common ancient name Hipparchos just meant horse-leader.) An implicit ancient comment on Athens' archons? Not that such an equation would retain the slightest applicability to the wise&pure leaders of today's astronomy-history institutions. Who, after decades of DR's calling them “archons”, may now recall the old (equally mature) joke: why did the Lone Ranger shoot Tonto? He found out “Kimo-Sabey” was Indian for poophead.]

Cantab John Couch Adams, double loser of the France-vs-England race to find Neptune (finished calculations 2nd, & no connexion to 1846/9/23 Berlin discovery), somewhat redeemed himself by swiftly using post-discovery Autumn observations in conjunction with J.Challis' pre-discovery 1846 Summer observations, to compute and announce (1846/10/15-17) the 1st nearly accurate orbit of Neptune.
However, note the CONTRAST between this speed — taking only 2 weeks to not only write his report but do the required calculations — versus the fact that he waited over 6 weeks after Cambridge learned (1846/10/1) of Neptune's 1846/9/23 Berlin discovery before producing (1846/11/13) his allegedly-already-computed-in-1845 “Hyp.1” orbital elements! This is one of the strangest of several evidences against Brit-mythology's long-accepted — now collapsing — claim of priority, entirely based upon that curiously reluctant “1845” orbit for Uranus' perturber. On his Dr.Cookian slowness to produce elements (not detailed calculations) that could have been immediately published, see: DIO 2.3 [1992] ‡9 §§D6&E3 [pp.127&128]; also DIO 9.1 [1999] ‡9 §H11 [p.21].
[North Pole faker Cook's furtive 1909 shiftiness regarding producing his records was amusingly and correctly weighed by W.Hobbs (D.Rawlins Peary at the North Pole: Fact or Fiction? 1973 pp.85&172). Cook's fellow-N.Pole faker Peary also delayed (1910/3/7-1911/1/7) in publicly producing his own records. But the key difference between Cook and either Adams or Peary is: Cook couldn't fake navigational data or use real data (which is harder), whereas Adams and Peary were genuinely expert at the relevant math of their endeavors — and so eventually produced data that for decades convinced onlookers that they had legitimate claims of priority.]

The bottle of bugs was half-full at 7:59.
[Almost no-one answers this question correctly — and most are educationally shocked at its powerful lesson: that exponential population-growth can sneak up on societies who aren't anticipating its sudden, explosive danger.]

On the twin-planets:
[a] Both pairs are of contiguous planets.
[b] The relative contiguity is closer than for other contiguous planets.
[c] Each pair's inner member rotates in retrograde.
[d] No other major planets do so.
[The foregoing set appears to constitute an original DIO discovery.]
These peculiarities may provide hints regarding the origin of the System. (DIO 1.1 [1991] ‡2 §G2 [p.14].)

Never. Venus' atmosphere refracts (bends) horizontal light-rays' paths far more than the curvature of Venus' sphere.
(DIO 2.3 [1992] ‡8 n.8 [p.100].)

Answer: Sunlight on Pluto is not dimmer but ordmag 1000 times brighter than full-moonlight on Earth.
(DIO 2.1 [1992] ‡1 §A3 [p.3].)
The full Moon is about 15 magnitudes dimmer than the Sun, i.e., roughly a million times dimmer. The Earth is about 31 times nearer the Sun than Pluto. Multiplying a millionth by the square of 31, we find a factor of about 1/1000.

Set p2 = n + 3. If p is integral, closed orbits will occur for F = krn. And p will provide the number of approaches to the gravitating center per revolution: obviously 1 for Newton (as we realize every January at Earth's perihelion) and 2 for Hooke (as is obvious from the 2-dimensional horizontal motion of a pendulum).
So for n = 6, 13, 22, 33, 46, etc, orbits will be closed.
(For n = −3, the orbiting body will steadily drift either towards or away from the central force. For values of n less than −3, the motion is unstable.)

ii is not imaginary or even complex. It is a real (if irrational) number. Hint: try Euler's Equation. Answer at our Defynitions page.

Most sources have listed Wm.Lassell as Umbriel's discoverer in 1851. However, DR and Chas.Kowal showed decades ago that Wm.Herschel (Uranus' 1781 discoverer) spotted Umbriel first, on 1801/4/17, and published his accurate positional data long before Lassell.
NOTE: By the time it was possible reliably to check Herschel's claim, Lassell's name had been attached for decades to Umbriel, thus creating an inertia that resembles the present difficulty astronomers are having in demoting Pluto from planethood. Though E.Bower and E.Brown warned immediately upon Pluto's 1930 discovery that Pluto was not the huge 7-Earth-mass planet P.Lowell had predicted in 1915, few noted or appreciated their simple logic: in the period between the last Uranian data (used by Lowell) and 1930, Uranus' path had followed normal ephemerides, not the motion entailed by the gravity of Lowell's large predicted body. Yet not until DR's widely-read 1968 March S&T article did the process of publicly facing Pluto's true puniness begin. But, as with the Umbriel situation, it was too late: for over a third of a century, Pluto had been wishfully or fallaciously deemed ordmag an Earth-mass, so it was naturally included in the list of planets. This has created traditions — even childhood-engendered loyalties — that will take a long time to overcome.
(One bright sign of hope on the Pluto front: see the excellent New York Times editorial of 2006/8/19.)

The Earth pulls on the Moon only half as hard as the Sun pulls on the Moon!
Because the Earth is roughly 1/300000 the Sun's mass (which is 400 times farther away from the Moon than is the Earth's mass), and 1/300000 divided by the inverse-square of 400 is about 1/2.
[If this seems strange, consider that: if the Earth vanished, the Moon would continue revolving around the Sun.]

Four. (Useful for patio-restaurants, where bothersome asymmetry is due more to floor than table.)
It's obvious that a 3-legged table will immediately find stability. But a 4-legged table just requires rotating until stability is achieved.
[This assumes that the table stands on a surface with a vertical gradient which is
[a] continuous, and
[b] of sufficient magnitude that one is not left with an impossible situation, e.g., the extreme case of two over-short diagonally-opposite legs on a table upon a perfectly planar floor.]

So that the man-hole cover cannot fall in the hole and hurt the man.

Answer: Uranus. Neptune hasn't been within 18AU of Pluto for millions of years, whereas Uranus was roughly 11AU from Pluto in recent decades. (Situation shown to be long-term-stable resonance in AJ 1969.)

A tossed rock is an artificial satellite — for a moment, anyway. But the general opinion that the motion is parabolic — a very close [flat-Earth] approximation to the truth — fails to comprehend that the correct viewpoint is that one is seeing the rock during just a tiny fraction of one sharpish end of the 6400 km long (and extremely eccentric) ELLIPTICAL orbit it would follow if it didn't crash into the ground.