Dinner for One

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Image: Wikimedia Commons

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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Image: Wikimedia Commons

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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Image: Wikimedia Commons

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

https://imgur.com/reddit-oc-topologists-map-of-world-map-showing-international-borders-nothing-else-qGVyb50

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

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Image: Wikimedia Commons

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.

O Loving Hate!

In 1988, Harvard mathematician Steven Strogatz hit on a new way to teach differential equations. Juliet loves Romeo, but Romeo is fickle: The more she warms to him, the colder he becomes, but when her interest wanes his feelings grow. She, on the other hand, tends to echo his feeling for her, returning love for love and hate for hate:

dr/dt = –aj, dj/dt = br,

where

r(t) = Romeo’s love/hate for Juliet at time t
j(t) = Juliet’s love/hate for Romeo at time t.

Positive values of r, j signify love, negative values signify hate, and the parameters a, b are positive, to be consistent with the story.

“The sad outcome of their affair is, of course, a neverending cycle of love and hate; their governing equations are those of a simple harmonic oscillator. At least they manage to achieve simultaneous love one-quarter of the time.”

The same idea can be pursued in other variations. He adds, “the term ‘many-body problem’ takes on new meaning in this context.”

(Steven H. Strogatz, “Love Affairs and Differential Equations,” Mathematics Magazine 61:1 [February 1988], 35.)

Dry

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It has been claimed that only three people read the three volumes of Russell and Whitehead’s monumental Principia Mathematica in its entirety, namely Russell, Whitehead, and the proofreader. There has been some scepticism, however, about Russell and Whitehead belonging to the list.

— Leo Moser, quoted in Howard Eves, Mathematical Circles Adieu, 1977

Polchinski’s Paradox

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Image: Wikimedia Commons

In 1990, physicist Joseph Polchinski described a puzzling situation: Suppose a billiard ball were sent through a traversable wormhole into the past, where it knocked its earlier self off course, away from the wormhole. This is a version of the famous grandfather paradox: How can we explain the ball’s presence where it seems to have prevented its own arrival?

Fortunately, Caltech students Fernando Echeverria and Gunnar Klinkhammer found that self-consistent solutions exist in which the older ball imparts only a glancing blow to its younger self, one that sends it through the hole at just the right angle to deliver a similarly glancing blow to its own forebear, and so on, sparing us from the paradox. (In 1991 the students and Kip Thorne reported that they’d failed to find any initial conditions in which such self-consistent solutions were unavailable, so it’s plausible that they exist for every possible initial trajectory. But this hasn’t been proven.)

Another pleasing possibility is that the older ball knocks the younger out of the way and itself enters the wormhole again. This creates an infinite loop, and some existential bewilderment, but no paradox.