Chernoff’s Faces

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:
Image: Wikimedia Commons

A Self-Enumerating Crossword

sallows crossword

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:


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]”.)

A Long Visit

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?

Click for Answer

Practical Math

Sample questions from L. Johnson’s 1864 textbook Elementary Arithmetic Designed for Beginners, used in North Carolina during the Civil War:

  1. A Confederate soldier captured 8 Yankees each day for 9 successive days; how many did he capture in all?
  2. If one Confederate soldier kill 90 Yankees how many Yankees can 10 Confederate soldiers kill?
  3. If one Confederate soldier can whip 7 Yankees, how many soldiers can whip 49 Yankees?

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.


night gallery

In the 1969 Night Gallery episode “Eyes,” Joan Crawford plays Claudia Menlo, a ruthless dowager who blackmails a doctor into performing a nerve transplant that will grant her vision for 11 hours. Afterward, alone in her apartment, she impatiently removes the bandages prematurely. She catches a glimpse of a crystal chandelier and then everything goes black. She rampages through her suite and collapses in tears, unaware that the city has suffered a power outage. At dawn, through dimming vision, she sees the rising sun, rushes to grasp it, and crashes through a window to her death.

Now: “How many things did Claudia Menlo see?” asks Dartmouth philosopher Roy Sorensen. “Most people say she saw only the chandelier and the sun (and possibly the pavement on the way down). But I say Claudia saw something in between seeing the crystal chandelier and the rising sun: the darkness of her blacked-out apartment. Claudia had never seen darkness before and mistook this visual experience for an absence of visual experience.”

Can we see darkness? Sorensen pictures a cave explorer in a completely dark cave. If the explorer is asleep and dreaming that he is in a completely dark cave, he does not see the darkness — but when he wakes up, he does. If the explorer then stands too quickly and the blood rushes from his head, he sees stars against an accurately perceived black background — the surrounding darkness. In contrast, his blind companion can’t tell whether the cave is dark; “only the sighted man can tell whether the cave is dark just by looking.”

This raises a puzzle: Suppose you’re in a light-tight container that’s suspended within a larger light-tight container. If the interior of the larger container is illuminated, then of course the darkness you see is the darkness of the smaller container. But what happens if the illumination of the larger space is turned off? You certainly can’t see beyond the walls of the small container in any circumstances. And only the larger container is blocking light. Does it follow that you’re seeing the darkness of the large container within the small container?

(Roy Sorensen, “We See in the Dark,” Noûs 38:3 [2004], 456-480.)