# Group Theory: Lagrange’s Theorem and the Sudoku Principle

Voltaire said to not let the perfect be the enemy of the good. In that spirit, I’ve decided to post a few (highly imperfect) video lectures I recently on group theory. There’s a lot of things I would do to go back and improve these lectures, but I hope that the content makes up for any flaws!

There will be a third one coming soon, and will be a working-through of an example that shows the power of Lagrange’s theorem. This format is a bit of an experiment for me, and I’ll keep working to make it better. I hope you enjoy!

# The Most Irrational of Numbers

Today I have a new interactive demonstration for you!

Controls
Zoom in: W
Zoom out: S
Reset angle: 0
Switch mode (auto or user): Space
On “User Mode”:
– Increase angle: Up
– Decrease angle: Down

[pjs4wp]
void setup() {
size(600,600);
background(0);
stroke(255);
fill(255);
}
float r = 2;
float da = 0;
float dda = .0001;
boolean autoPlay = true;
void draw() {
background(0);
text(“Angle = ” + (float)((int)(da*10000))/100. + “%”, width/2, 20);
text(“Zoom = ” + (float)((int)(r*100))/100., width/2, 35);
if (autoPlay)
text(“Mode: Auto”, width/2, 50);
else
text(“Mode: User”, width/2, 50);
translate(width/2,height/2);
int i = 0;
float angle = 0;
while (width/r > i) {
float x = r*i*cos(angle);
float y = r*i*sin(angle);
ellipse(x,y,5,5);
angle += 2*PI*da;
i++;
}
if (keyPressed) {
if (key == ‘w’)
r *= 1.01;
if (key == ‘s’)
r *= .99;
if (key == ‘0’)
da = 0;
if (!autoPlay) {
if (keyCode == UP)
da += dda;
if (keyCode == DOWN)
da -= dda;
}
}
if (autoPlay)
da += dda;
}
void keyReleased() {
if (key == ‘ ‘)
autoPlay = 1 – autoPlay;
}
[/pjs4wp]

(As always, zooming out too much will make things begin to lag. Also, fair warning, things start getting a little seizure-inducing around a zoom of .5 or less.)

Okay! Pretty, right? But what’s actually going on?

Each frame we are drawing a sequence of dots. We start at the center of the screen, and take constant radial steps outwards. And for each step, we apply some fixed angular rotation. This angular rotation is given as a percentage of 2π on the top of the screen, and is the thing being adjusted frame by frame (either automatically or based on your input, depending on which mode you choose).

What’s cool is what happens when you approach simple rational numbers. Take a look for yourself by going on “User” mode and bringing the angle to 10% (for 1/10), 14.28% (for 1/7), 25% (for 1/4), and so on. Here are two examples:

Cool, right? At every rational number, we end up with a number of “spokes” corresponding to the denominator of our rational number. People sometimes describe the golden ratio Ф as the “most irrational number”, meaning that in some sense it is the most difficult to approximate by rational numbers. How does this look on our visualization? Check it out!

And zooming out further, we see…

We can see that using Ф as our angular rotation gives us the “least spokey” shape possible. As we zoom out further and further, we will always see our points looking basically evenly distributed across the angles at any given radius. This is a wonderfully visual way to understand the irrationality of Ф!

I recently came across a whole bunch of crazy historical trivia, involving the laws around adultery, BDSM, and divorce. Here are some of the quotes that made me gasp (mostly from Wikipedia):

As of 2019, adultery remains a criminal offense in 19 states, but prosecutions are rare. Although adultery laws are mostly found in the conservative states (especially Southern states), there are some notable exceptions such as New York, Idaho, Oklahoma, Michigan, and Wisconsin consider adultery a felony, while in the other states it is a misdemeanor.

Penalties vary from a \$10 fine (Maryland) to four years in prison (Michigan). In South Carolina, the fine for adultery is up to \$500 and/or imprisonment for no more than one year (South Carolina code 16-15-60), and South Carolina divorce laws deny alimony to the adulterous spouse.

In Florida adultery (“Living in open adultery”, Art 798.01) is illegal; while cohabitation of unmarried couples was decriminalized in 2016.

Under South Carolina law adultery involves either “the living together and carnal intercourse with each other” or, if those involved do not live together “habitual carnal intercourse with each other” which is more difficult to prove.

In Alabama “A person commits adultery when he engages in sexual intercourse with another person who is not his spouse and lives in cohabitation with that other person when he or that other person is married.”

In some Native American cultures, severe penalties could be imposed on an adulterous wife by her husband. In many instances she was made to endure a bodily mutilation which would, in the mind of the aggrieved husband, prevent her from ever being a temptation to other men again. Among the Aztecs, wives caught in adultery were occasionally impaled, although the more usual punishment was to be stoned to death.

The Code of Hammurabi, a well-preserved Babylonian law code of ancient Mesopotamia, dating back to about 1772 BC, provided drowning as punishment for adultery.

Amputation of the nose – rhinotomy – was a punishment for adultery among many civilizations, including ancient India, ancient Egypt, among Greeks and Romans, and in Byzantium and among the Arabs.

In England and its successor states, it has been high treason to engage in adultery with the King’s wife, his eldest son’s wife and his eldest unmarried daughter. The jurist Sir William Blackstone writes that “the plain intention of this law is to guard the Blood Royal from any suspicion of bastardy, whereby the succession to the Crown might be rendered dubious.”

Adultery was a serious issue when it came to succession to the crown. Philip IV of France had all three of his daughters-in-law imprisoned, two (Margaret of Burgundy and Blanche of Burgundy) on the grounds of adultery and the third (Joan of Burgundy) for being aware of their adulterous behaviour. The two brothers accused of being lovers of the king’s daughters-in-law were executed immediately after being arrested.

Until 2018, in Indian law, adultery was defined as sex between a man and a woman without the consent of the woman’s husband. The man was prosecutable and could be sentenced for up to five years (even if he himself was unmarried) whereas the married woman cannot be jailed.

In Southwest Asia, adultery has attracted severe sanctions, including death penalty. In some places, such as Saudi Arabia, the method of punishment for adultery is stoning to death. Proving adultery under Muslim law can be a very difficult task as it requires the accuser to produce four eyewitnesses to the act of sexual intercourse, each of whom should have a good reputation for truthfulness and honesty. The criminal standards do not apply in the application of social and family consequences of adultery, where the standards of proof are not as exacting.

Adultery is no longer a crime in any European country. Among the last Western European countries to repeal their laws were Italy (1969), Malta (1973), Luxembourg (1974), France (1975), Spain (1978), Portugal (1982), Greece (1983), Belgium (1987), Switzerland (1989), and Austria (1997).

In most Communist countries adultery was not a crime. Romania was an exception, where adultery was a crime until 2006, though the crime of adultery had a narrow definition, excluding situations where the other spouse encouraged the act or when the act happened at a time the couple was living separate and apart; and in practice prosecutions were extremely rare.

English common law defined the crime of seduction as a felony committed “when a male person induced an unmarried female of previously chaste character to engage in an act of sexual intercourse on a promise of marriage.” A father had the right to maintain an action for the seduction of his daughter (or the enticement of a son who left home), since this deprived him of services or earnings.

In more modern times, Frank Sinatra was charged in New Jersey in 1938 with seduction, having enticed a woman “of good repute to engage in sexual intercourse with him upon his promise of marriage. The charges were dropped when it was discovered that the woman was already married.”

Buddhist Pali texts narrate legends where the Buddha explains the karmic consequences of adultery. For example, states Robert Goldman, one such story is of Thera Soreyya. Buddha states in the Soreyya story that “men who commit adultery suffer hell for hundreds of thousands of years after rebirth, then are reborn a hundred successive times as women on earth, must earn merit by “utter devotion to their husbands” in these lives, before they can be reborn again as men to pursue a monastic life and liberation from samsara.

According to Muhammad, an unmarried person who commits adultery or fornication is punished by flogging 100 times; a married person will then be stoned to death. A survey conducted by the Pew Research Center found support for stoning as a punishment for adultery mostly in Arab countries; it was supported in Egypt (82% of respondents in favor of the punishment) and Jordan (70% in favor), as well as Pakistan (82% favor), whereas in Nigeria (56% in favor) and in Indonesia (42% in favor) opinion is more divided, perhaps due to diverging traditions and differing interpretations of Sharia.

The Roman Lex Julia, Lex Iulia de Adulteriis Coercendis (17 BC), punished adultery with banishment. The two guilty parties were sent to different islands (“dummodo in diversas insulas relegentur”), and part of their property was confiscated. Fathers were permitted to kill daughters and their partners in adultery. Husbands could kill the partners under certain circumstances and were required to divorce adulterous wives.

Durex’s Global Sex Survey found that worldwide 22% of people surveyed admitted to have had extramarital sex. In the United States Alfred Kinsey found in his studies that 50% of males and 26% of females had extramarital sex at least once during their lifetime. Depending on studies, it was estimated that 26–50% of men and 21–38% of women, or 22.7% of men and 11.6% of women, had extramarital sex. Other authors say that between 20% and 25% of Americans had sex with someone other than their spouse. Three 1990s studies in the United States, using nationally representative samples, have found that about 10–15% of women and 20–25% of men admitted to having engaged in extramarital sex.

The Standard Cross-Cultural Sample described the occurrence of extramarital sex by gender in over 50 pre-industrial cultures. The occurrence of extramarital sex by men is described as “universal” in 6 cultures, “moderate” in 29 cultures, “occasional” in 6 cultures, and “uncommon” in 10 cultures. The occurrence of extramarital sex by women is described as “universal” in 6 cultures, “moderate” in 23 cultures, “occasional” in 9 cultures, and “uncommon” in 15 cultures.

Traditionally, many cultures, particularly Latin American ones, had strong double standards regarding male and female adultery, with the latter being seen as a much more serious violation.

Adultery involving a married woman and a man other than her husband was considered a very serious crime. In 1707, English Lord Chief Justice John Holt stated that a man having sexual relations with another man’s wife was “the highest invasion of property” and claimed, in regard to the aggrieved husband, that “a man cannot receive a higher provocation” (in a case of murder or manslaughter).

The Encyclopedia of Diderot & d’Alembert, Vol. 1 (1751), also equated adultery to theft writing that, “adultery is, after homicide, the most punishable of all crimes, because it is the most cruel of all thefts, and an outrage capable of inciting murders and the most deplorable excesses.”

# On BDSM

The United States Federal law does not list a specific criminal determination for consensual BDSM acts. Some states specifically address the idea of “consent to BDSM acts” within their assault laws, such as the state of New Jersey, which defines “simple assault” to be “a disorderly persons offense unless committed in a fight or scuffle entered into by mutual consent, in which case it is a petty disorderly persons offense”.

Mutual combat, a term commonly used in United States courts, occurs when two individuals intentionally and consensually engage in a fair fight, while not hurting bystanders or damaging property. There is not an official law that forbids mutual combat in the United States. There have been numerous cases where this concept was successfully used in defense of the accused. In some cases, mutual combat may nevertheless result in killings.

Oregon Ballot Measure 9 was a ballot measure in the U.S. state of Oregon in 1992, concerning sadism, masochism, gay rights, pedophilia, and public education, that drew widespread national attention. It would have added the following text to the Oregon Constitution:

All governments in Oregon may not use their monies or properties to promote, encourage or facilitate homosexuality, pedophilia, sadism or masochism. All levels of government, including public education systems, must assist in setting a standard for Oregon’s youth which recognizes that these behaviors are abnormal, wrong, unnatural and perverse and they are to be discouraged and avoided.

Dildos or any object used for “the stimulation of human genital organs” cannot be made or sold in Alabama. The Anti-Obscenity Enforcement Act says that anyone caught with such tools could face a fine up to \$20,000, a one-year jail sentence or 12-months doing hard labor.

Florida bans “lewd and lascivious behavior,” which is defined as a situation where “any man and woman, not being married to each other, lewdly and lasciviously associate and cohabit together.” The misdemeanor is punishable by a fine of up to \$500. In Mississippi, an unmarried couple caught living together “whether in adultery or fornication” can face up to six months in jail and/or a \$500 fine.

In 2003, the U.S. Supreme Court deemed a Texas state law that banned the practice of anal and oral sex between same-sex couples as unconstitutional. Despite the ruling, a sizable list of states, including Texas, still have anti-sodomy laws on the books.

Louisiana’s “crime against nature” statute prohibits the “the unnatural carnal copulation by a human being with another of the same sex or opposite sex or with an animal.” The state legislature in April failed to pass a bill that would have repealed the law except for human-on-animal relations.

Other states that have some form of anti-sodomy laws include Kansas, Oklahoma, Alabama, Florida, Idaho, Louisiana, Mississippi, North Carolina, South Carolina, and Utah, according to the Human Rights Campaign. Virginia repealed its ban in March.

# On Divorce

Today, every state plus the District of Columbia permits no-fault divorce, though requirements for obtaining a no-fault divorce vary. California was the first U.S. state to pass a no-fault divorce law. Its law was signed by Governor Ronald Reagan, a divorced and remarried former movie actor, and came into effect in 1970. New York was the last state to pass a no-fault divorce law; that law was passed in 2010.

Prior to the advent of no-fault divorce, a divorce was processed through the adversarial system as a civil action, meaning that a divorce could be obtained only through a showing of fault of one (and only one) of the parties in a marriage. This required that one spouse plead that the other had committed adultery, abandonment, felony, or other similarly culpable acts. However, the other spouse could plead a variety of defenses, like recrimination (essentially an accusation of “so did you”). A judge could find that the respondent had not committed the alleged act or the judge could accept the defense of recrimination and find both spouses at fault for the dysfunctional nature of their marriage. Either of these two findings was sufficient to defeat an action for divorce, which meant that the parties remained married.

Before no-fault divorce was available, spouses seeking divorce would often allege false grounds for divorce. Removing the incentive to perjure was one motivation for the no-fault movement.

In the States of Wisconsin, Oregon, Washington, Nevada, Nebraska, Montana, Missouri, Minnesota, Michigan, Kentucky, Kansas, Iowa, Indiana, Hawaii, Florida, Colorado and California, a person seeking a divorce is not permitted to allege a fault-based ground (e.g. adultery, abandonment or cruelty).

In some states, requirements were even more stringent. For instance, under its original (1819) constitution, Alabama required not only the consent of a court of chancery for a divorce (and only “in cases provided for by law”), but equally that of two-thirds of both houses of the state legislature. This requirement was dropped in 1861, when the state adopted a new constitution at the outset of the American Civil War. The required vote in this case was even stricter than that required to overturn the governor’s veto in Alabama, which required only a simple majority of both houses of the General Assembly.

These requirements could be problematic if both spouses were at fault or if neither spouse had committed a legally culpable act but both spouses desired a divorce by mutual consent. Lawyers began to advise their clients on how to create legal fictions to bypass the statutory requirements. One method popular in New York was referred to as “collusive adultery”, in which both sides deliberately agreed that the wife would come home at a certain time and discover her husband committing adultery with a “mistress” obtained for the occasion. The wife would then falsely swear to a carefully tailored version of these facts in court (thereby committing perjury). The husband would admit a similar version of those facts. The judge would convict the husband of adultery, and the couple could be divorced. Specifically, they report that “states that adopted no-fault divorce experienced a decrease of 8 to 16 percent in wives’ suicide rates and a 30 percent decline in domestic violence.”

The Code of Hammurabi (1754 BC) declares that a man must provide sustenance to a woman who has borne him children, so that she can raise them:

If a man wish to separate from a woman who has borne him children, or from his wife who has borne him children: then he shall give that wife her dowry, and a part of the usufruct of field, garden, and property, so that she can rear her children. When she has brought up her children, a portion of all that is given to the children, equal as that of one son, shall be given to her. She may then marry the man of her heart.

In the 1970s, the United States Supreme Court ruled against gender bias in alimony awards and, according to the U.S. Census Bureau, the percentage of alimony recipients who are male rose from 2.4% in 2001 to 3.6% in 2006. In states like Massachusetts and Louisiana, the salaries of new spouses may be used in determining the alimony paid to the previous partners.

Some of the possible factors that bear on the amount and duration of the support are:

 Factor Description Length of the marriage or civil union Generally, alimony lasts for a term or period. However, it will last longer if the marriage or civil union lasted longer. A marriage or civil union of over 10 years is often a candidate for permanent alimony. Time separated while still married In some U.S. states, separation is a triggering event, recognized as the end of the term of the marriage. Other U.S. states do not recognize separation or legal separation. In a state not recognizing separation, a 2-year marriage followed by an 8-year separation will generally be treated like a 10-year marriage. Age of the parties at the time of the divorce Generally, more youthful spouses are considered to be more able to ‘get on’ with their lives, and therefore thought to require shorter periods of support. Relative income of the parties In U.S. states that recognize a right of the spouses to live ‘according to the means to which they have become accustomed’, alimony attempts to adjust the incomes of the spouses so that they are able to approximate, as best possible, their prior lifestyle. Future financial prospects of the parties A spouse who is going to realize significant income in the future is likely to have to pay higher alimony than one who is not. Health of the parties Poor health goes towards need, and potentially an inability to support oneself. The courts are disinclined to leave one party indigent. Fault in marital breakdown In U.S. states where fault is recognized, fault can significantly affect alimony, increasing, reducing or even nullifying it. Many U.S. states are ‘no-fault‘ states, where one does not have to show fault to get divorced. No-fault divorce spares the spouses the acrimony of the ‘fault’ processes, and closes the eyes of the court to any and all improper spousal behavior. In Georgia, however, a person who has an affair that causes the divorce is not entitled to alimony.