In 1917, W.S. Andrews inscribed a sphere with five circles of latitude and six of longitude.
Then he arranged the numbers 1 to 62 on the intersections such that each of the 11 circles bears 12 numbers totaling 378.
A Magic Globe
Science & Math
Speed Limit
We think of light speed as dizzyingly fast, but it’s still disastrously slow given the roomy universe we find ourselves in. NASA astronomer James O’Donoghue designed these videos to illustrate. If we could bend it around Earth, light would orbit the planet in 0.13 seconds:
And it takes only 1.255 seconds to reach the moon:
But it takes fully 3 minutes to reach our neighboring planet Mars at its closest approach:
Light from the sun takes more than 8 minutes to reach Earth. To reach the closest star takes more than 4 years.
Still Life
The programming language efghij is literally object-oriented — its logic is expressed by arranging ordinary items in meaningful tableaus:
In this example, the tray and the cutting board constitute code blocks. The oblong plate, however, has a clothes peg on it, which expresses a for loop. The fork is a local variable. The fork variable is initialized to 1 within the for loop’s initial-value expression. The sauce with the rubber band indicates that the number of iterations is equal to the function’s argument. The stuff on the cutting board is the loop body. Using a stapler, the fork is multiplied by the sauce and the result assigned to the fork using a drinking glass. A hammer decrements the sauce, which demonstrates that the function’s parameter can be used as a local variable too. Finally, the toilet roll indicates that the return value is the final value of the fork.
That produces a factorial. The program below outputs the phrase “Hello, world!” — it’s just a pillow bearing a piece of paper on which those words are written.
Fare Division
There are four states in which matter can exist: solid, liquid, gas, and plasma. Similarly, in an April Fools’ Day paper in 2022, Harvard’s Madelyn Leembruggen and Caroline Martin proposed that foods can be classified as soup, salad, or sandwich:
- “Soups are a suspension of ingredients in a liquid matrix, whereas salads are a suspension of ingredients in a gaseous matrix.” Thus cereal is soup, and mixed nuts are salad. The transition between phases is continuous: Depending on how finely they’re chopped, the same set of ingredients might produce a chunky salad, a salsa, or a gazpacho. Vanilla ice cream is a “glassy” soup because it doesn’t flow at its serving temperature, but it takes on a more liquid character as it melts.
- A soup or salad becomes a sandwich when it’s enclosed in a carbohydrate. The shape and quality of the carbohydrate must vary according to the substance it encloses, but a proposed general rule is that if you can pick up a dish by the carb and eat it, it’s a sandwich.
Further investigation is needed into some edge cases (e.g., lasagna vs. baked ziti), but in most cases a lot of ground can be covered by just asking “Is this a soup, a salad, or a sandwich?” “Most of the time, the answer you blurt out with the least amount of thought is the most true.”
“Finally, to remove any of the reader’s lingering uncertainty: yes, a hotdog is a sandwich.”
(Madelyn Leembruggen and Caroline Martin, “What’s for Lunch? A Systematic Ordering of Foods in the Soup-Salad-Sandwich Phase Space,” arXiv preprint arXiv:2203.16580 [2022].)
Dinner for One
In humans, each eye has a small blind spot corresponding to the point where the optic nerve leaves the retina. Normally this isn’t noticeable because the brain can fill in the gap with information from the other eye. But by closing one eye and positioning the target at the appropriate angle, it’s possible to invoke the gap deliberately.
“When the famous physiological psychologist Karl S. Lashley was forced to endure an irritating dinner guest, he amused himself by bringing his blind spot over the person’s head, neatly decapitating the person.”
— Jearl Walker, The Flying Circus of Physics, 2006
Synopsis
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Mazes are designed to be tortuous, so even a published solution can be hard to follow. A “straight-line diagram” dispenses with the windings and converts the essential puzzle in a simple map that’s easy to interpret:
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This example summarizes the hedge maze in Barcelona’s Parc del Laberint d’Horta.
The same can be done with any meandering route — many subways have adopted stylized maps that dispense with geography and focus on the order of stops, the most important detail for most commuters.
Small World
Reddit user xilefakamot created this in 2020 — a “topologist’s map of the world.”
It depicts international borders and nothing else — the shapes, sizes, and distances between countries are forsaken in order to present the pattern of their borders as efficiently as possible.
“No calculations per se – I sketched out a series of networks showing which countries bordered each other, then gradually smoothed them out to make the map – all done manually.”
(Via MapPorn.)
A Bug’s Life
A bicycle has wheels of different sizes. On the rim of each wheel is a bug. As the bike travels, the bug on the smaller wheel traces a large number of small arches, and the bug on the larger wheel traces a smaller number of large arches. On a given bicycle trip, which bug travels the greater distance through space?
Remarkably, they travel the same distance. The arches are cycloids, and one revolution of a wheel produces one cycloid arch of length 8r, where r is the wheel’s radius. With each revolution the wheel travels a distance 2πr along the road. This means that during a bicycle journey of distance D, the wheel makes D/2πr revolutions, and the attending bug’s total distance through space is (D/2πr)(8r) = 4D/π. The rs have dropped out, which means that the size of the wheels doesn’t matter — the length of each bug’s path through space is 4/π times the length of the bicycle journey.
Via Doug Rohrer’s 1993 book Thought Provokers.
Somewhat similar: How much paint does my carousel need?
Applied Geometry
George Wither wrote this “rhomboidal dirge” in 1622:
Ah me!
Am I the swain
That late from sorrow free
Did all the cares on earth disdain?
And still untouched, as at some safer games,
Played with the burning coals of love, and beauty's flames?
Was't I could dive, and sound each passion's secret depth at will?
And from those huge o'erwhelmings rise, by help of reason still?
And am I now, O heavens! for trying this in vain,
So sunk that I shall never rise again?
Then let despair set sorrow's string,
For strains that doleful be;
And I will sing,
Ah me!
But why,
O fatal time,
Dost thou constrain that I
Should perish in my youth's sweet prime?
I, but awhile ago, (you cruel powers!)
In spite of fortune, cropped contentment's sweetest flowers,
And yet unscornèd, serve a gentle nymph, the fairest she,
That ever was beloved of man, or eyes did ever see!
Yea, one whose tender heart would rue for my distress;
Yet I, poor I! must perish ne'ertheless.
And (which much more augments my care)
Unmoanèd I must die,
And no man e'er
Know why.
Thy leave,
My dying song,
Yet take, ere grief bereave
The breath which I enjoy too long,
Tell thou that fair one this: my soul prefers
Her love above my life; and that I died her's:
And let him be, for evermore, to her remembrance dear,
Who loved the very thought of her whilst he remained here.
And now farewell! thou place of my unhappy birth,
Where once I breathed the sweetest air on earth;
Since me my wonted joys forsake,
And all my trust deceive;
Of all I take
My leave.
Farewell!
Sweet groves, to you!
You hills, that highest dwell;
And all you humble vales, adieu!
You wanton brooks, and solitary rocks,
My dear companions all! and you, my tender flocks!
Farewell my pipe, and all those pleasing songs, whose moving strains
Delighted once the fairest nymphs that dance upon the plains!
You discontents, whose deep and over-deadly smart
Have, without pity, broke the truest heart.
Sighs, tears, and every sad annoy,
That erst did with me dwell,
And all other joys,
Farewell!
Adieu!
Fair shepherdesses!
Let garlands of sad yew
Adorn your dainty golden tresses.
I, that loved you, and often with my quill,
Made music that delighted fountain, grove, and hill;
I, whom you loved so, and with a sweet and chaste embrace.
Yea, with a thousand rather favours, would vouchsafe to grace,
I now must leave you all alone, of love to plain;
And never pipe, nor never sing again!
I must, for evermore, be gone;
And therefore bid I you,
And every one,
Adieu!
I die!
For, oh! I feel
Death's horrors drawing nigh,
And all this frame of nature reel.
My hopeless heart, despairing of relief,
Sinks underneath the heavy weight of saddest grief;
Which hath so ruthless torn, so racked, so tortured every vein,
All comfort comes too late to have it ever cured again.
My swimming head begins to dance death's giddy round;
A shuddering chillness doth each sense confound;
Benumbed is my cold sweating brow
A dimness shuts my eye.
And now, oh! now,
I die!
In 1968, conceptual artist Douglas Huebler used small bits of self-sticking paper to mark the vertices of two giant imaginary hexagons in Boston and New York. “The world is full of objects, more or less interesting; I do not wish to add any more,” he wrote later. “I prefer, simply, to state the existence of things in terms of time and place.”
Harmony
By CMG Lee, a pleasing visual proof that the number of possible handshakes among n people is the (n-1)th triangular number.
The first pair shake hands, and each new arrival shakes with everyone already present. The cumulative total counts every possible handshake.