If an equilateral triangle is inscribed in, and has a common vertex with, a rectangle, as shown above, then areas A + B = C.
If a triangle with angles α, β, γ is inscribed in, and has a common vertex with, a rectangle, as shown below, and if the right triangles opposite α, β, γ have areas A, B, C, respectively, then A cot α + B cot β = C cot γ.
Somewhat related: A Curious Equality.
(Tom M. Apostol and Mamikon Mnatsakanian, “Triangles in Rectangles,” Math Horizons 5:3 [February 1998], 29-31.)
On Sept. 26, 1901, 13-year-old Fleetwood Lindley was attending school in Springfield, Ill., when his teacher handed him a note: His father wanted him urgently. He rode his bicycle to the Oak Ridge cemetery two miles out of town and found his father, Joseph, in the memorial hall of Abraham Lincoln’s tomb. The assassinated president, now 36 years dead, was being transferred to a new resting place, and a small group of caretakers had decided to open his coffin to confirm his identity.
The casket had been laid across a pair of sawhorses. A pair of workmen used a blowtorch to unseal the lead panel that covered Lincoln’s upper body, and the small group peered in.
Afterward the coffin was lowered into a hole 10 feet deep, encased in a cage of steel bars, and buried under tons of concrete. Over the years, as the other witnesses passed away, Lindley became the last living person to have looked on Lincoln’s body.
“His face was chalky white,” he remembered for a Life reporter in 1963, three days before his own death. “His clothes were mildewed. And I was allowed to hold one of the leather straps as we lowered the casket for the concrete to be poured.”
“I was not scared at the time, but I slept with Lincoln for the next six months.”
In 1983, Soviet satellites reported that the United States had launched a nuclear missile toward Moscow, and one officer had only minutes to decide whether to initiate a counterstrike. In today’s show we’ll learn about some nuclear near misses from the Cold War that came to light only decades after they occurred.
We’ll also hear listeners’ input about crescent moons and newcomers to India, and puzzle over the fatal consequences of a man’s departure from his job.
operose
adj. involving great labor
(Thanks, Trevor.)
In walking through a park and observing the signs, ‘All dogs found on these grounds without their owners, will be shot,’ a friend of mine exclaimed, ‘That’s a hard case for dogs that can’t read.’
— Irving Browne, Humorous Phases of the Law, 1876
Humans are bad at evaluating complex data, but we’re good at reading faces. So in 1973 Stanford statistician Herman Chernoff proposed using cartoon faces to encode information. He found that up to 18 different data dimensions can be represented in a computer-drawn face, mapping one variable to the length of the nose, another to the space between the eyes or the position of the mouth, and so on. This produces an array of faces that we can assess quickly using the brain’s natural talent for reading features. (The example above shows lawyers’ ratings of state judges in U.S. Superior Court.)
“This approach is an amusing reversal of a common one in artificial intelligence,” Chernoff noted. “Instead of using machines to discriminate between human faces by reducing them to numbers, we discriminate between numbers by using the machine to do the brute labor of drawing faces and leaving the intelligence to the humans, who are still more flexible and clever.”
(Herman Chernoff, “The Use of Faces to Represent Points in K-Dimensional Space Graphically,” Journal of the American Statistical Association 68:342 [June 1973], 361-368.)
Nepal’s constitution contains complete instructions for drawing its flag:
(A) Method of Making the Shape Inside the Border
(1) On the lower portion of a crimson cloth draw a line AB of the required length from left to right.
(2) From A draw a line AC perpendicular to AB making AC equal to AB plus one third AB. From AC mark off D making line AD equal to line AB. Join BD.
(3) From BD mark off E making BE equal to AB.
(4) Touching E draw a line FG, starting from the point F on line AC, parallel to AB to the right hand-side. Mark off FG equal to AB.
(5) Join CG.(B) Method of Making the Moon
(6) From AB mark off AH making AH equal to one-fourth of line AB and starting from H draw a line HI parallel to line AC touching line CG at point I.
(7) Bisect CF at J and draw a line JK parallel to AB touching CG at point K.
(8) Let L be the point where lines JK and HI cut one another.
(9) Join JG.
(10) Let M be the point where line JG and HI cut one another.
(11) With centre M and with a distance shortest from M to BD mark off N on the lower portion of line HI.
(12) Touching M and starting from O, a point on AC, draw a line from left to right parallel to AB.
(13) With centre L and radius LN draw a semi-circle on the lower portion and let P and Q be the points where it touches the line OM respectively.
(14) With centre M and radius MQ draw a semi-circle on the lower portion touching P and Q.
(15) With centre N and radius NM draw an arc touching PNQ [sic] at R and S. Join RS. Let T be the point where RS and HI cut one another.
(16) With Centre T and radius TS draw a semi-circle on the upper portion of PNQ touching it at two points.
(17) With centre T and radius TM draw an arc on the upper portion of PNQ touching at two points.
(18) Eight equal and similar triangles of the moon are to be made in the space lying inside the semi-circle of No. (16) and outside the arc of No. (17) of this Schedule.(C) Method of Making the Sun
(19) Bisect line AF at U and draw a line UV parallel to line AB touching line BE at V.
(20) With centre W, the point where HI and UV cut one another and radius MN draw a circle.
(21) With centre W and radius LN draw a circle
(22) Twelve equal and similar triangles of the sun are to be made in the space enclosed by the circles of No. (20) and of No. (21) with the two apexes of two triangles touching line HI.(D) Method of Making the Border
(23) The width of the border will be equal to the width TN. This will be of deep blue colour and will be provided on all the sides of the flag. However, on the five angles of the flag the external angles will be equal to the internal angles.
(24) The above mentioned border will be provided if the flag is to be used with a rope. On the other hand, if it is to be hoisted on a pole, the hole on the border on the side AC can be extended according to requirements.Explanation: The lines HI, RS, FE, ED, JG, OQ, JK and UV are imaginary. Similarly, the external and internal circles of the sun and the other arcs except the crescent moon are also imaginary. These are not shown on the flag.
That’s a good thing — it’s the only national flag that’s not a quadrilateral. The two pennants represent different branches of a ruling dynasty in the 19th century. The nation signaled its pride in the new design last February by setting a world record for the largest human flag — 35,000 Nepalese gathered in Kathmandu to break Pakistan’s record and to demonstrate their own national unity. I wonder how they worked out the geometry:
Here’s a unique crossword puzzle by Lee Sallows. There are no clues — instead, each of the 12 entries must take the form [NUMBER](space)[LETTER](S), like so:
EIGHT BS
NINETEEN XS
ONE J
And so on. Can you complete the puzzle so that the finished grid presents an inventory of its own contents?
(A couple observations to get you started: Because the puzzle contains 12 entries, the solution will use only 12 letters. And one useful place to start is the shortest “down” entry, which is too short to be plural — it must be “ONE [LETTER]”.)
Your tedious nephews, Kerry and Kelly, are not honest, but they’re orderly. One of them lies Mondays, Tuesdays, and Wednesdays, and tells the truth on other days, and the other lies on Thursdays, Fridays, and Saturdays, and tells the truth on other days. At noon, they have the following conversation:
Kerry: I lie on Saturdays.
Kelly: I will lie tomorrow.
Kerry: I lie on Sundays.
On which day of the week does this conversation take place?
Sample questions from L. Johnson’s 1864 textbook Elementary Arithmetic Designed for Beginners, used in North Carolina during the Civil War:
Students were also asked to imagine rolling cannonballs out of their bedrooms and dividing Confederate soldiers into squads and companies. Let’s hope they didn’t take field trips.
