The Unit Circle

Truth to tell, I was disappointed to find a really nice write-up on Wikipedia.  I was hoping that this was still arcane knowledge.  As far as I can tell, it is not commonly taught in trigonometry classes.  We use the sine and cosine for problems with vectors; and that seems to be about it.  In point of fact, these constructions are the root and rock of computational trigonometry.  If you draw well and measure carefully, you can get two decimal places, or fractions to eighths.

Words have meaning.
If you ever suffered from a respiratory allergy,
then you know that it affects the "bay" the "sinus" cavities.
What we call the sine of an angle is the half-sine,
the semi-bay.
Similarly, our "tangent"
is the measure of the semi-tangent,
ignoring the reflection
below the part that we care about.

The trigonometric identities
come from the Pythagorean theorem.

In Feynman's Lost Lecture,  the professor allows that we do our maths with algebra and calculus.  We no longer rely on geometric constructions. Feynman had to create his own derivations for Newton's Laws simply because he could not follow Newton's easy claims about conic sections.  Newton used geometry to create the calculus.  However, calculus is such a powerful tool that we stopped learning the geometry that Newton knew. 

You can write out the algebraic statements
but a picture is worth a thousand words.

In Cosmos, Carl Sagan tells that Pythagoras fled from Samos because he could not tolerate the tyrant Polycrates, whom Sagan denigrates for having "started out as something like a caterer."  (Bold though he was, Sagan shared the anti-capitalist mentality.) But in The Ancient Engineers, de Camp tells us that engineers working for Polycrates bored through a mountain, starting at opposite ends, met in the middle, and were not off by a foot in a mile.  Lacking the positional notation of Arabic numerals, of course, they did all their calculations with geometry.

Descartes is credited with uniting algebra and geometry. 
His work reflects the seemingly intractable 
analytic-synthetic dichotomy of philosophers. 
In truth, it should have erased the distinction 
between the logically consistent and the empirically verifiable.
See the works of Gregory Browne on this blog here and here.

Long ago, at Curious Books in East Lansing, I found a old manual for apprentice carpenters that showed sines, cosines, and tangents to 32nds of an inch for triangles of given measurement.  Many times I have regretted not buying it then.  Even before that, when Laurel and I were first married, her father asked us what calculus was all about. We told him.  He asked, "Do you mean like this?" and took out a foot-sized caliper and showed us how he measured stair cases.  Rise over Run.  Ain't no doubt.

PREVIOUSLY ON NECESSARY FACTS
Pi in the Sky over Austin (2014)
Patent Nonsense
World Peace Through Massive Retaliation
Circumference
Anthropocene: A Bad Name for a Good Thing

An Abundance of Talent: the 2015 Austin Energy Regional Science Festival

For the fourth time, I served as a judge for behavioral and social sciences in our local science fair.  Again, I met an array of talented and motivated teenagers.  They were intelligent (of course), actively curious about the world around them, willing to step out from the crowd and put themselves in the scales to be judged.  They asked interesting questions and pursued the answers wherever the data took them.  But they were, after all, children.  Some of them assumed far too easily that an experiment that does not validate the hypothesis is therefore a failure.  They never heard of Karl Popper.  That failing is not theirs, but of their mentors – or the lack of them. 

Middle school (junior high) presentation
on noise levels in the school building

 I am pleased and proud to have argued for the first place winner in senior high school  behavioral and social sciences. (See all of the Awards here.) One of the judges said that when he challenged her on a point of mathematics, she did not have the answer.  I responded with some history: last year, she asked every one of the mathematics and science teachers in her high school for help with statistics and she got no replies to her emails. So, she went to the university; and some UT doctoral candidates tutored her in statistics.  So, too, this year, did she seek and find outside help in order to extend and expand her work in statistical methods.  Personally, I was the one who was challenged.  I got an A-minus in my undergraduate class in statistics.  After reading her abstract this year, I downloaded several tutorials: she knew more than I did.

 

Middle School enquiry into which advertising
message draws the most responses

We expect a lot from kids. The German word for “teenager” is Halbstark: half-strong.  That speaks to the core of the problem in a way that the Latin “adolescent” (becoming adult) does not.  My daughter had a mole on her wrist; and she would show me how it moved around as she grew.  For them, life is an intense process.  We judge them as if they were adults.  As a geometer would say, it is obvious by inspection that they are not. Yet, objectively, nothing less is fair to them in the intellectual pursuit of science.

 

Classical, country, dub step, meditation, or pop:
does any help you concentrate on a task?

“If you all were graded on a 100-point scale, 91 would be failing.”  As often as I said it, I could see that it did not sink in, not this year, not in the previous years. This year, I asked one panicked entrant if any other display was clearly head and shoulders better than hers.  The person with the neighboring display chimed in: “The right answer is ‘No.’” 

 

Middle School entry: What is Your "Pawsonality"?
Can a psychological profile predict your preference for a pet?

It is not just kids at science fairs.  I enter and I judge museum quality exhibits at numismatic conventions.  (“Four out of five? How dare they!”)  In the West Wing episodes that bridge the first and second seasons, President Jed Bartlett says that decisions are made by those who show up.  In this context, the future of science, engineering, and technology belongs to – and will be claimed by – those who enter the competitive field of scientific research.

 

Another middle school entry on the Stroop Effect.
This took second place.
The value in this for the learner is figuring out how to
create a novel experiment and enter it in a competition.

The best of them do it alone; but they all deserve mentoring.  That 9-point gap between first and last could easily be closed by a working technician, engineer, or scientist who made the time to volunteer with a school starting in August or September. It is not a matter of showing them how, but of asking science-talented pupils those tough questions early on. 

PREVIOUSLY ON NECESSARY FACTS

ELI the ICE Man

The Structure of Scientific Revolutions

The Austin Energy Regional Science Festival 2014

Teaching Science Through Science Fiction

The Problem of Cultural Patrimony

From Agent Presse France 
via Al-Arabiya online here 

On February 18, 2015, news services (Reuters here) reported that a group of amateur undersea divers from Israel found 2000 gold coins from about 1000 CE, off the coast of the ancient city of Caesarea which is now in the modern state called Israel.  The coins were struck during  the Fatimid Caliphate.  Speculation by archaeologists from the Israeli Antiquities Authority included the suggestion that the coins were taxes collected for the ruling center of the time at Cairo in Egypt.  An alternate theory is that the coins came from a private merchant ship moving from port to port.  No article cited the problem of patrimony.

Fatimid dinars feature the names of the caliphs they were minted under, as well as the date and location where they were minted. "They're first-class historical documents," explains Robert Kool, curator of the IAA's Coin Department.  […] A cursory study reveals that the earliest coin from the hoard was minted in Palermo, Sicily, while the majority came from official Fatimid mints in Egypt and other parts of North Africa and date to the reigns of Caliphs al-Hakim (A.D. 996-1021) and his son al-Zahir (A.D. 1021-1036).  -- National Geographic News online here.

UNESCO (the United Nations Educational, Scientific, and Cultural Organization) has a broad mandate.  In the past fifteen years, it has arranged with national governments around the world – chief among them, the United States of America – for the passage of laws requiring that “cultural patrimony” be returned to the country of origin.  The Boston Museum of Fine Art and the Getty Museum are among the most newsworthy victims. Several problems haunt the laws. The problem of patrimony is multidimensional. The claims of UNESCO and their allies suffer from internal contradictions because they have no objective basis in epistemology: no standard of value exists for identifying right from wrong.

First, modern nations did not exist in ancient times.  The Boston Museum returned to Turkey a statue called “The Weary Herakles” from 200 BCE. But there was no “Turkey” in 200 BCE.   They called it “Helliades” – and you need to understand what that means.

In the Iliad and the Odyssey, the Greeks called themselves “Achaeans.”  About 400 years later, during the Hellenistic Era, they called themselves “Hellenes.”  Hellas is the modern Greek name for Greece.  But grammatically Helliades is much wider than Hellinike. 

Hellenike would be any Greek city with ties to the old homeland.  Tarentum in Italy was Taras; Naples (Napoli) was Neopolis.  Marseilles in France was Marsalla.  Benghazi in Libya was Berenike.  They were Greek cities.  Greeks built them; Greeks inhabited them.  Sicilians are not Italians.  Italy is a peninsula. Sicily is an island. They are different places.  The people of eastern Sicily are Greeks. In the west, they are Carthaginian.

However, after Alexander the Great, many cities of diverse ethnicities came into the Greek sphere. The orator Demosthenes said, “The name ‘Hellene’ no long applies to a race, but to a state of mind.”

 

Ancient Coin Collectors Guild (here)
lobbies for the rights of private citizens

On that basis, should not every coin from Pakistan to Spain, from Egypt to England go back to Athens? In truth, as the laws are applied, coins from Greek cities of Sicily do not go back to the government museums of Greece: they go to the government museums of Italy. 

Consider the Roman Empire.  In addition to the central mint, coins were struck in what are today Tunisia, Egypt, Turkey, Bulgaria, Croatia, Germany, France, and England.   Moreover, Roman legions came from all across the empire, but could be stationed anywhere.  The legion name (cognomen) could reflect where they were drawn from or where they won a battle.  We have epigraphic evidence to show that Legio IX Hispanica served in England from 65 to 120 CE. Legio III Cyrenica (modern Libya) served in what are today Egypt, Syria, and Iraq. Legio III Italica was stationed in what is Regensburg for 200 years, from 165 CE.  Galba reorganized former marines into Legio I Adiutrix (“Rescuer”) in 68 AD, and stationed them in what is today Mainz.  However, for most of the second century (106 to 198) their base was in a town later known for many years as Szőny in Hungary, but which is now called Komárom on the border with the Slovak Republic.  So, when a hoard of coins found in England contains the product of a mint from Croatia to pay Roman troops from Spain, whose patrimony is it?

Second, the assumption of UNESCO and the co-operating national governments is that the national public museums have a moral right to first claim.  Private collectors are rebuked as looters of protected sites. 

Old World traditions acknowledge the ruler as the owner of all property. The king could grant a title and land to go with it because all of his realm was his to dispose of as he saw fit.  In America today, that continues as eminent domain: the state can take what it needs.  So, when a farmer in Egypt or England finds an ancient coin on his land, it is not his at all: it belongs to the national government.  (England relaxed its “treasure trove” laws; but even that granting of latitude was from the rulers down to their subjects.) The governments of Turkey, Greece, Italy claim that these artifacts were "looted" from "historic sites" but it is easy to make any place historic if other people ever lived there before you did.  Historicity is just an excuse.  In their minds, private property does not exist.

“I’ve been reading only about the Italians and Greeks 

and how they’ve succeeded,’’ said [former Turkish 

cultural official Engin] Ozgen.  “This will show the world 

that the Turks are not ignorant anymore, that they will fight 

for their past and their heritage.’’ This statue is attributed 
as a Roman copy of a lost Greek work.  
It comes from the city of Antalya in modern Turkey.
The town was founded by Attalus II of Pergamon in 180 BCE. 

 Seljuk Turks entered the area about 1500 years later. 

The Boston Museum of Fine Art returned this 
statue of "The Weary Herakles" to the city of Antalya. 

Read here.

Third, why is a museum in Rome more entitled to these artifacts than I am?  It is my patrimony as well.  And, like most Americans, I have a lot of patrimony.  In addition to the Sicilians (Greeks), my mother’s side of the family was loosely identified as “Hungarian.” However, my maternal grandmother’s maiden name was Croatian.  The family just moved up the river inside of a large, multi-ethnic empire in central Europe.  My maternal grandfather’s immigration records identify him as German-Hungarian.  Are not the artifacts of the legions also my cultural inheritance?

Fourth and more to the point, I cite Demosthenes above: “Hellene” is a state of mind.  UNESCO and the national governments couch their claims in the language of race, ethnicity, and genetics.  The fact of the matter is that we are creatures of intellect living in a planetary society.  Seiji Ozawa (Japanese; born in China) and Yo-Yo Ma (Chinese; born in Paris) have as much right to buy Beethoven’s belt buckle as do any of the current inhabitants of Bonn.  

Fifth, national borders are fluid.  Economist Ludwig von Mises (1881-1973) and his brother, mathematician Richard von Mises (1883-1953), were born in a town called Lemberg in the Austro-Hungarian empire.  Treaty of Versailles put the town in the Ukraine, where today it is called L’viv.  According to the UNESCO theory of culture, the people there lost their patrimonic right to enjoy the artifacts of one milieu, but were given the consolation prize of another matrix of icons and exemplars. 

Coin of Sinope in modern Turkey, struck about 375 BCE 


when Diogenes the Cynic was mint master.  


Accused of debasing the coinage, he moved to Athens 

and became a philosopher. 

Under the eagle's wing DIO below the dolphin, SINO. 

The test cut shows that the coin was suspected 
of being false.

The ANA measured the specific gravity and found it 

to be only 
90% pure silver, acceptable to us, 


but debased to the ancients.

Ultimately, the ideological nationalization of museums contradicts the fundamental intention of a museum.  Every leading institution seeks to own and display as broad a collection as possible. That motivation derives from the liberal Enlightenment theory of humanity.  Growing up in Cleveland, Ohio, I benefited from time spent at the Cleveland Museum of Art. It seemed to me then – and is still supportable today – that the entirety of human culture was housed there. 

A hundred yards away, the Natural Science Museum displayed meteorites and dinosaur fossils.  Those, too, are arguably someone else’s patrimony. One of the displays about Humans was a panel explaining Culture. I do not know if the Jivaro would be culturally enhanced by the return of their shrunken heads, or if the Inuit need another old canoe; but I certainly am better for having studied them without trekking to Peru and Alaska.

ALSO ON NECESSARY FACTS

Circumference

Numismatics: History as Market

Newton versus the Counterfeiter

Accounting for Civilization

Turn Left at Orion

As with perhaps all enduring efforts, this book was written for the author himself. Even though – or perhaps because - Guy Consolmagno had worked as a postdoctoral researcher in planetary astronomy, and taught at Harvard and MIT, he had no appreciation for what a small telescope could do. 

Turn Left at Orion 

by Guy Consolmagno and Dan M. Davis, 

Cambridge University Press, 

1989, 1995, 2000, 2011; 

reprinted with corrections 2012; 

4th Edition 4th Printing, 2014.

As he tells it in the Introduction, he had quit his job and signed on with the Peace Corps. His friend, Dan M. Davis, was enthusiastic about buying him a small telescope to take to Africa. Consolmagno was doubtful. After all, what can you see with a 3-inch refractor? Within the glare of New York City (Fort Lee, New Jersey), Davis showed Consolmagno the star Albireo, a stunning double star - one yellow, the other blue. (It is at the head of the Swan or the foot of the Cross, designated β Cygni in the catalogues.)

The other problem that Consolmagno had was with the instrumentation: setting circles, gauges, ascension, declination,… When he approached stellar astronomy as an amateur, he found all of the standard books unhelpful.  So, he wrote Turn Left at Orion.

“Every year, thousands of telescopes are sold, used once or twice to look at the Moon, and then they end up gathering dust in the attic. It’s not that people aren’t interested – but on any given night there may be 2,000 stars visible to the naked eye, and 1,900 of them are pretty boring to look at through a small telescope. You have to know where to look …”

This book tells you how to find them by following visual directions based on the obvious constellations and asterisms. (An asterism is a piece of a constellation. What we commonly call the Big Dipper is really the middle of the Great Bear, Ursa Major. The Sword of Orion and the Belt of Orion are also common asterisms.)

Amateur astronomy has changed since Consolmagno was a boy in Detroit in the 1950s and 60s. Even in his childhood, the hobby had come far in the previous 50 years. A hundred years ago, 1914, the typical amateur telescope was a 3-inch refractor housed in extensible brass tubes. By the 1960s such scopes were Christmas presents for kids; and reflectors of 8 inches (200 mm) were the mainstream for serious amateurs. John Dobson shifted the paradigm with his homemade telescopes. (Read Dobson’s biography on Wikipedia here.)

Turn Left at Orion rates your viewing 1 through 4 with silhouette images for icons: Dobsonian, Refractor (also for reflector; both are called “catadioptric”), and Binoculars. Dobsonian, catadioptric, and all of that are also explained in clear text with large illustrations, from the tripod to the mathematics of telescopes and their eyepieces.

This book is about Deep Sky Objects. It does deliver 25 pages of good information up front about the Moon and Planets. But they are easy enough to find. Successfully meeting the challenges makes a hobby fun.

Major sections of the book are organized by the seasons in the northern hemisphere: January-March; April-June; July-September; October-December. Within those are important asides about diffuse nebulae, open clusters, planetary nebula, galaxies, and globular clusters. Thirty-two pages are dedicated to the southern skies, from Hawaii on down to Chile and New Zealand. You have to be very far south to see the so-called “Clouds of Magellan”, smaller galaxies within our own group. 

“Consolmagno attended the University of Detroit Jesuit High School before he obtained his B.A. (1974) and M.A. (1975) degrees at Massachusetts Institute of Technology and his Ph.D. (1978) at the University of Arizona's Lunar and Planetary Laboratory, all in planetary science. After postdoctoral research and teaching at Harvard College Observatory and MIT, in 1983 he joined the US Peace Corps to serve in Kenya for two years, teaching astronomy and physics. After his return he took a position as Assistant Professor at Lafayette College in Easton, Pennsylvania.

“In 1989 he entered the Society of Jesus, and took vows as a brother in 1991. On entry into the order, he was assigned as an astronomer to the Vatican Observatory, where he also serves as curator of the Vatican Meteorite collection, positions he has held since then. In addition to his continuing professional work in planetary science, he has also studied philosophy and theology.” -- Guy Consolmagno – Wikipedia here.

The overall intention of this book is to provide you with a friend who will be your guide to the stars. That was the role that Dan M. Davis filled for Guy Consolmagno.

(Originally posted to The International Astronomy Forum December 15, 2014. Edited and updated 11 November 2020.)

ALSO ON NECESSARY FACTS

Seeing in the Dark

The Perfect Machine

Burnham's Celestial Handbook

Astronomical Symbols on Ancient and Medieval Coins

Great Bean Chocolate

We often meet vendors at the Wheatsville Co-op and Whole Foods, but never have we seen one so busy.  It was hard to catch Great Bean Chocolate without a couple or three people at her table.  

"We never use any questionable products in our bars; no soy, no GMO, no preservatives, and no gluten. Why so picky? Well, Great Bean aims to deliver the most delectable, nutrient-dense, sustainably grown, and fairly traded cacao on the face of the earth. It all boils down to an even simpler mission: to spread happiness & well-being through chocolate."

We bought the "Love" bar with figs, cinnamon, and love herbs: yohimbe, damiana, and maca root.

Previously on Necessary Facts
Hail Merry Desserts
Awesome Austin Foods
Valentine's Day: Love and Money

The Hat Creek Pickle Company

The Hat Creek Burger Company is a local chain of fast food restaurants.  One evening, I met one of the local coin club guys at Recycled Reads, the retail used book store of the Austin Public Library.  On our way to the coin club meeting we stopped at Hat Creek Burgers; and it was okay.  This afternoon, I met their latest venture, Hat Creek Pickles, at the Wheatsville Coop.  

Hat Creek Burger Co.,
5902 Bee Caves Road, West Lake Hills, Texas

After trying five kinds, I bought the Texas Sauerkraut.  Writing this, I am on my second serving.  It is spicy but balanced, crispy and crunchy.   It is two dollars a jar more expensive, but dollars being so small these days, the marginal value is easy to perceive.  Sauerkraut is high in vitamin C.  A day without sauerkraut is like a day without sunshine. 

Not Drew Gressett

• No artificial preservatives, calcium chloride, or other chemical firming agents added

• Free of sugar & corn syrup

• Supports immune system

• Probiotic and rich in enzymes

• Lactose & gluten free

• Increases the quantity, availability, digestibility and assimilation of nutrients in the body

(I note here that those claims are not substantiated by the Food and Drug Administration – as opposed to claims about Viagra and Ritalin, which are approved by the FDA.  You have to wonder what the dialog would be like if you could bring Thomas Jefferson and Alexander Hamilton forward to our time.)

"In addition to brewing and bottling beer, Drew Gressett, owner of Hat Creek Burger Co., plans to use Strangeland Brewery to ferment pickles for the newly formed Hat Creek Pickle Co.  “A pickle makes a hamburger,” Gressett said. “To take the time to make our own pickle is synonymous with the care all our food receives.” – Impact News (Austin Metro) 

Previously on NecessaryFacts

Awesome Austin Foods at the Wheatsville Co-op

High Brew Cold Coffee

Formulas for Success

Shooting the Roadhouse Relics Gallery

Variations on Enigma

Verifying secret information is a challenge.  After enjoying the movie Imitation Game, I went to the library for books on Bletchley Park and Enigma. I found Enigma by Wladislaw Kozaczuk. Kozaczuk tells a story completely different from the one offered by F. W Winterbotham in The Ultra Secret.

I had read Andrew Hodges’ biography, Alan Turing: the Enigma back in the early nineties.  I am pretty sure that I read F. W. Winterbotham’s The Ultra Secret fifteen years earlier because it was sold by Loompanics Unlimited, who paid me to write The Code Book: All About Unbreakable Codes and How to Use Them. (On NecessaryFacts here.)

Discussing Imitation Game online, I read from others several twice-told tales that just were not so. I wrote several myself before I read Kozaczuk’s book.  I now regard it as the correct version – if the truth can have “versions.”

It is true that the Polish device for mechanically attacking the Enigma ciphers was called a “bomba.”  It was named for a large spheroid chocolate cake.  It was not shaped like the cake at all.  According to their story, it was only what they were having for dessert at a restaurant when they discussed the code name for their project.

 “The bomb was an electro-mechanical aggregate based on six Polish Enigmas, combined with additional devices and transmissions.  An electrically driven system of rotors revolved automatically, creating in each bomb, successively, over a period of about two hours (100-120 minutes), 17,576 different combinations. When the rotors aligned with the sought-for position, a light went on, the motors stopped automatically, and the cryptologist read the indications.   Thus, by setting in motion the bombs (six were built at once in November 1938), the daily keys could be recovered within two hours. “ (page 53)

 Mathematicians working for Polish intelligence had been attacking Enigma with some success for over ten years.  They had their own Enigma machines.  Built by reverse-engineering, they were variations on the commercial model. That was essentially the same foundation as the German devices that had been modified for military use.

Mathematician Marian Rejewski developed a theoretical model of the German Enigma. His work achieved realization in the construction of the Bomba.  That device depended on electronic elements created by AVA Radio Manufacturing, a small firm in Warsaw which had been working for their government since the firm was launched in the winter 1929/1930.

The story of AVA in particular and the Polish cryptanalysis project in general should be familiar to Americans today.  One of the founders of AVA never completed school beyond the fourth grade, but started out as an apprentice locksmith.  The company’s key to government work came via another founder who worked in the Cipher Bureau.  At heart, AVA was a small shop run by two brothers who were amateur radio operators.  After high school (“gymnasium” in Europe) they built and sold cheap little radio receivers.  But they were pretty good at their craft.

Also diligent were the young mathematicians from Poznan University: Marian Rejewsky, Henryk Zygalski, and Jerzy Rozycki.  (The book has a pronunciation guide up front:  Rejewski is “Rey EF ski”.)  Recruited specifically to work on German communications, Rejewski spent a year in Göttingen studying advanced statistics.  It was his judgment that Germany’s mathematics programs had fallen behind, and were not producing leading edge ideas.  Moreover, as would be borne out later, the German military did not trust mathematicians because they would get lost in theory instead of producing immediate results.  From our viewpoint today, that choice was evidence of the failure of their philosophical assumptions. 

By 1934, the Polish mathematicians and radio hams had collaborated to build four Enigma duplicates.

None of that appeared in Winterbotham’s book. Neither did Alan Turing have a place in the book.  In the movie, Imitation Game, Commander Alastair Denniston was portrayed as a martinet of limited vision.  Unfair as that may well have been, it is recorded independently (or at least repeated often), that Denniston considered Enigma to be unbreakable.  It is hard to find success at something you regard as impossible.  Nonetheless, Winterbotham gave Commander Denniston full credit for running the group—headed by Alfred Dillwyn “Dilly” Knox, who received one mention only—that broke Enigma.   

In fact, Winterbotham said of Knox that he was “quite young” when working at Bletchley Park.  Knox was born in 1884. He had been with the UK’s “black chamber” in World War I; and he is credited with delivering the decryption of the Zimmerman Telegram.

PREVIOUSLY ON NECESSARY FACTS

A Successful Imitation of Alan Turing

Astronomical Symbols on Ancient and Medieval Coins

The Sokal Affair

Reflections on the Sokal Affair

Science versus Common Sense