Handwritten letter on multiple pages of thin but rough unlined 6½" x 9½" paper, sent to my uncle Dobe while he was in prison. About ⅔ of the letter is from Peter; the rest from Marcy. The Peter section includes a couple of hand-drawn math/geometry things, including a supposed method for trisecting any angle with compass and straight edge, so I’m including more images than usual in this letter.
In the trisection section, for simplicity, I’m using the arc symbol (⌒) before a set of letters to indicate an arc, in place of the more standard notation that Peter used (drawing an arc above the letters).
I feel like I should mention before you get to it that the puzzle about where to build the bridge is a trick question, or at least a think-outside-the-box question. Other than that, my notes are at the end of this post as usual.
20 Jan. 1969
dear john--
hope this finds you well & happy--your last letter sounded like you’re in pretty good spirits…
BIG NEWS! jedediah has TWO new teeth, for a total of FIVE! he now has:
Diagram of Jed’s teeth.
x’s, and will get ☮ before he stops popping them for a while… he fell off a bed 2 days ago & learned about edges & falling--with a cut on eyebrow for souvenir… veteran’s badge…
here’s the only answer i’ve known to your poser:
Five rooms puzzle, with part of the path drawn on top of one of the walls.
, with path from B to A going “across” a line in a different sense/way than the other crossings…
o, here’s a neat one: a lame farmer wants to build a bridge from house H to barn B, across river R, such that the edges of the bridge are square (⟂) to the river’s edges:
Diagram for bridge puzzle, showing H and B on opposite sides of river R, with some distance between them along the riverbank, and a possible bridge location between them.
question: where should he build the bridge so as to have the shortest walking distance?
answer: oho, you were going to peek! well, i’ll fold up the answer on a separate paper, don’t look until you think about it a while…
Written sideways in the margin: i might have a job as science-teacher’s aide in high school… also took a test to work at phone co.…
now, the moment you’ve been waiting for, 2000 yrs in show-biz & never until now a TRISECTION of anx arbitrary angle, using only compass & straight-edge: in fact, even better, an obvious extension of this idea will yield an N-SECTION of any angle! beauteous, noumenous, wondrous… discovered by PLASTIC-MAN who invented the fiery-whooshing mandala technique:
Diagram showing a way to drip burning plastic from a can.
LIGHT THE STRAW, WATCH LIGHTS ON CEILING, ENJOY WHOOSHINGS OF FALLING FLAME!
anyway, that’s how he got his name, xxx then he was bus-driver for 6-day school, has a kid named RJB (pronounced “ARJBY”), now he’s gone & discovered fabled trisection method, drew it up on 4'x8' plywood & exhibited it on haight st., sent proof in to scientific american! HERE IT IS! (ready or not…)
to trisect ∠AOB:
Diagram showing how to trisect an angle.
first bisect, obtain OC: then draw ⌒DEF, with center at O: then ⌒GH, center at E, (arbitrary radiixx for ⌒DF & ⌒GH), then mark I at distance R from H & describe ⌒HJ, center at I, and K, distance R from G, & describe ⌒GL, center at K. Now draw construct line through L ∥ to AO, and [construct] the [line through] J ∥ to BO: these lines meet at Q on CO: draw GQ & HQ. NOW, ∠LQJ = ∠AOB (rt.side ∥ rt. side, left side ∥ left side): xx ⌒LKG = ⌒GEH = ⌒HIJ, and ∠LQG = ∠GQH = ∠HQJ! since ∠LQJ = ∠Q is trisected, & ∠Q = ∠AOB = ∠O, this completes the trisection! however, if you want to transfer this back to O, draw the parallel to GQ through O & the parallel to HQ through O, then ∠AOM = ∠MON = ∠NOB!
HURRAY! HUZZAH! it’s so simple, it’s like you perform the trisection on a shadow-plane (dotted lines), then project back to the plane it’s a shadow of (solid lines)! (xxx all circles arcs shown xxx should be dotted for you to see it more clearly…) (can extend this to pentasecton, or heptasection, or N-section!)
all love——
peter, marcy, jed
This section is in Marcy’s handwriting.
Dear Brother John
Peter says I should add the wish that you not become embroiled in nostalgia—sit around & sing a New Song, take a sad song & make it better
Hope you can work out all that geometry. It’s so beautiful xxx & so simple I hardly believe it.
I’ve been learning morse code — beginning, anyway — feel strongly that it will be needed some day, wld like to get a ham license. (I’ve always suspected that ham in ham stood for something, like heavy amateur media-tor or some such, but I guess not.)
Jedediah the Toothsome is bouncing on his Daddy’s shoulders. Did we ever tell you that Jedidah was the Name the prophet Nathan gave to Solomon before said king changed it to Solomon—
Small lord wants some nursin’ — have to go — much love from us all,
Marcy
The rest of the letter is in Peter’s handwriting again.
Solution to bridge puzzle: a very wide bridge.
(he builds a wide bridge, as shown…)
(ho, ho, ho)
tarot-astro-kabal class going very well, thank you: we made & colored a large ‘tree’ diagram, nice to have around, do you have paints?
did you notice that proportions of christmas card were in golden ratio? also the oval, also itxs position…
also:
distance from pate to navel:
[distance from] navel to sole::
[distance from navel to sole]:
[distance from pate to sole]!
a proof for you to work on if you want:
Diagram illustrating question of how to prove that the intersections of a triangle’s trisections form an equilateral triangle.
Given: △ABC, each ∠ trisected, trisectors meet in D, E, F
Prove: △DEF is equilateral
(i don’t know the proof, but it’s true, no tricks…)
fold a thin piece of paper in a knot like this
Drawing showing tying a strip of paper into a knot.
(overhand knot) & HOLD UP TO LIGHT…
Notes
“your poser”
This is known as the five-room puzzle. That Wikipedia article shows a neat animation of a solution to the puzzle if it’s on a torus, but if it’s on an ordinary flat sheet of paper, it’s impossible to solve without cheating in some way, like what Peter did in his answer here.
burning dripping plastic
A friend told me about something similar in the ’90s; I don’t know whether it was just circulating in the community all that time, or developed independently, or what.
This definitely isn’t a correct method of exactly trisecting an angle; it has been proved that that’s not possible to do. I think I know what’s wrong with it, but I welcome discussion of same. I wouldn’t be surprised if this turns out to be a well-known false trisection, but a quick web search didn’t turn anything up.
Marcy learning Morse Code
This is the first I’ve heard of Marcy having any interest in Morse or in radio operation. Peter had been, I think, a ham radio operator in his teen years; and he set up a radio (with a very big wire loop for an antenna) at our house years later. But I didn’t know Marcy had ever been interested in that.
etymology of ham
Wikipedia indicates that the term derives from professional radio operators pejoratively referring to amateurs as ham-fisted. I don’t know whether that’s true, though, and my dictionary doesn’t help.
my name
Marcy misspelled Jedidiah here. And as far as I can tell, Solomon was first named Solomon, and then Nathan said to call him Jedidiah; for more, see a post of mine from 2014.
“distance from pate to navel”
In case my formatting of those lines isn’t clear, Peter was saying that the ratio of two distances in the human body is equal to the ratio of two other distances. I assume that, as with most such notions, this is a vast overgeneralization and approximation; human bodies vary in many ways.
I still do this sometimes with strips of paper like the paper that holds a straw or a pair of chopsticks. If you tie such a knot and flatten it out, the central shape is a pentagon.
