A roundup of the best holiday sweets and snacks for your festivities. by Martha Daghlian

Throughout centuries of human civilization, people have sought out ways to combat the darkness and cold temperatures of winter: from ancient temples that aligned with the rising solstice sun to the ceremonial burning of symbolic fir trees, we have devised many strategies to brighten up the longest season. But perhaps the best solution to the winter doldrums is to have a little treat? Cultures all around the world have invented their own special cookies, cakes, and sweets to be enjoyed at this time of year, and we’ve rounded up some of our faves that are available right here in Portland!

Fancy Cake Thinks it’s a Tree

Arguably the most iconic of Christmas desserts, the bûche de noël (or yule log) predates current “is it cake” fads by well over a century. Traditionally made with light sponge cake rolled around a rich filling and adorned with cute meringue mushrooms, this whimsical faux tree echoes the ancient Celtic tradition of burning a ceremonial log at Winter Solstice and really amps up the classic holiday aesthetic. Don’t have a spare 57 hours and a background in French pastry? Let the professionals at Pix Patisserie take care of all your bûche needs with one of their glossy high concept stumps, available to order through December. Pix Patisserie, 2225 E Burnside, pixpatisserie.com

Baklava URCU ATALAY TANKUT / GETTY IMAGES

Baklava Just Like
Grandma Used to Make

Within the Armenian-American side of my family, baklava—a buttery, syrupy nut-and-phyllo pastry popular across the Mediterranean, Central/West Asian, and North African regions—is a mandatory holiday treat. I grew up on my grandmother’s recipe, which calls for walnuts only (no pistachios, thank you), an entire pound of clarified butter, NO cinnamon (how dare you even suggest it!), and a sneaky splash of bourbon, presumably a modern twist added by grandma. But TBH, I haven’t met a baklava I didn’t love, and there are a few particularly strong ready-made contenders here in Portland: World Foods Market, with locations in the Pearl and on Barbur Boulevard, makes a few varieties of baklava and similar pastries, all of which are exquisite (even with pistachios). 

Sophisticated Citrus

It’s a serendipitous fact that winter is not only a season for feasting on rich foods, it’s also the time of year when bright, refreshing citrus is at its peak. Who doesn’t love a bowl of oranges at a holiday party? In addition to enjoying fresh citrus fruit straight up (my favorite is the dekopon or sumo mandarin), there are all sorts of fancy things you can make with the help of our zesty friends: fresh grapefruit mimosas, traditional pomander balls (that’s when you poke a ton of little holes in an orange and shove a whole dried clove in each one, creating a spicy little air freshener that also kind of looks like a medieval weapon as it slowly dries up throughout the winter), hot toddies with tons of lemon. If you really want to impress your friends, try making your own candied citrus peel—it’s like a grown up version of those fruit slice gummy candies! 

Candied Citrus Peel

Cut the peel from a bunch of citrus fruits (make sure not to include any of the white part!) into strips. Blanch them three times (that means placing them in a saucepan, covering with water, and boiling for five minutes, then you’ll drain and boil them for five more minutes in fresh water, then do the same thing one more time). Then, cover them with a 1:1 mixture of water and granulated sugar. Bring to a boil, then simmer for 40-60 minutes over low heat. They should become soft and a little translucent at the edges. Drain and place on a drying rack until cool (put some foil or parchment underneath to catch any drips). Once they are dry, you can toss them in sugar for a sparkly and crunchy coating.

Candied citrus peel makes a super classy garnish for fancy desserts, and can also be used in cake and muffin recipes. Or just snack on them in between all the cookies for a “healthy” treat—it’s fruit, isn’t it??

Sticky Sweets for Lunar New Year

Lunar New Year may still be a little ways off, but I’m already planning what I’ll bring to the annual party our friends throw to celebrate this traditional East Asian holiday. A lot of the foods associated with Lunar New Year celebrations symbolize some specific kind of luck that one might hope to attract in the coming months. Noodles, dumplings, fish, and citrus all connect in some way to ideas including longevity, wealth, and unity. But in my humble opinion, treats featuring sticky rice are the star of the show. This time around, I plan to visit Li Min Bakery at 81st and SE Division, and Shop Halo Halo on 50th and SE Woodstock, for traditional nian gao (sticky rice cakes) and moon cakes filled with sweet bean paste. Lin Min’s Bakery & Bistro, 8615 SE Division; Shop Halo Halo, 4981 SE Woodstock, STE 2, shophalohalo.com

Rugelach Nataly Hanin / Getty Images

Rad Rugelach

My first encounter with rugelach was at an elementary-school friend’s house, where her mom taught us to make this classic holiday cookie, originally dreamed up hundreds of years ago by Jewish bakers in Poland. Wait, is it a cookie? Or is it a pastry? You know what, I don’t care… all I know is it comes in lots of different flavors (Apricot! Raspberry! Chocolate!) and its twisty little crescent-moon shape really spices up a cookie plate. Like many delicious wintertime snacks, they do take a bit of effort to make at home, so if you’re short on time you might want to check out Henry Higgins Bagels, which contrary to their name, also serves up rugelach, babka, and challah on the regular. Henry Higgins Boiled Bagels, multiple locations, hhboiledbagels.com

Scandinavian-Style Snacks

If this list of festive treats still isn’t enough for you, mark your calendars for the 40th annual Scan Fair! This massive event, hosted by Nordic Northwest at the Oregon Convention Center December 7 and 8, is inspired by traditional Scandinavian Christmas markets. Think traditional singing and dancing, cozy knitwear, and tons of delicious sweets, snacks, and drinks–basically a massive party to brighten up the dark wintertime with some serious hygge. My most trusted Scandinavian treat advisor suggests loading up on Æbleskiver (little round pancakes, often served with lingonberry jam), pickled herring, and glögg (spiced wine punch). Scan Fair runs Dec. 7-8 at the Oregon Convention Center, 777 NE MLK Jr, get your tickets and find more info at nordicnorthwest.org/scanfair 

Today, Tufts University announced that Loring Tu, a professor in the Department of Mathematics, has made a substantial gift to name the Science and Engineering Complex (SEC) in honor of his late grandfather, Tsungming Tu, who was a world-renowned doctor and expert on pharmacology, toxicology, and medical education. Going forward, the SEC will be known as the Tsungming Tu Complex (TTC).

Private equity is more likely to look at public firms for acquisition when the target aggressively manages its earnings to meet analyst expectations, because those firms are often so cut to the bone there's more room to add value with investment.

Today, Tufts University announced that Loring Tu, a professor in the Department of Mathematics, has made a substantial gift to name the Science and Engineering Complex (SEC) in honor of his late grandfather, Tsungming Tu, who was a world-renowned doctor and expert on pharmacology, toxicology, and medical education. Going forward, the SEC will be known as the Tsungming Tu Complex (TTC).

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Content Warning: Incredibly ugly upcoming U.S. Politics. Stephen Miller and Donald Trump's public promise for a "bloody story" - plans for sweeping raids and mass deportations of 10 to 20 million people living in America are expected to start when Trump assumes office and begins on day one.

People living all across the United States, from farm camps in the backwaters of Florida to the ethnically diverse neighborhoods of major cities will likely be greviously impacted. Currently it is expected that the combined utilization of the U.S. Immigration and Customs Enforcement (ICE), National Guard, local police, trained and armed civilians and possibly even the US Military will take part in the raids to capture legal and illegal immigrants. Sympathetic police departments will have their funding slashed if they don't participate in the raids. Surprisingly, some research suggests a majority of Americans support mass deportations and concentration camps, which was reflected in the election results. Whether the popular support continues after it begins depends on the effectiveness of ICE and the Department of Homeland Security's indoctrination and propaganda programs that they have already been preparing for this moment, as well as the mass media's whitewashing of the proceedings as an "affordable housing program," among other things. So the question is, what can we realistically expect? ICE's "Citizens Academy" program offers up some clues...

The academy trains civilians to operate multiple firearms, use lethal force, perform surveillance on immigrants, and conduct raids while also acting as a public relations initiative to try and sway public opinion about ICE, Customs and Border Protection (CBP), and the HSI unit of the Department of Homeland Security (DHS). "We ran Border Patrol Citizen Academies back when I was an agent as well," Jenn Budd, former Senior Border Patrol Agent, whistleblower, and author of Against the Wall told Unicorn Riot. "The entire goal is to indoctrinate locals into why it's okay for us to violate people's rights." The training materials that show where to strike people with batons to subdue them are but the tip of the iceberg in terms of violence. While the ICE presentation on using force contains the "Monadnock Baton Chart" (presumably named after baton manufacturer Monadnock) showing the degrees of damage inflicted on a suspect based on where they're struck with a baton, the majority of this part of Citizens Academy training covers when to use deadly force.

What could happen to the people swept up? The program is generally portrayed as mass deportations, but already it has become clear that many countries (even Mexico) are expected not to welcome the tidal wave of migrants. Instead, the prison-industrial complex which is being flooded with investment money is gearing up to build large scale prisons and camps to house them. Many of these migrants can be expected to be kept there for years on end, with the Supreme Court supporting indefinite detetion of immigrants, an affirmation of long standing U.S. policy. Slavery in America is still legal in the prison system (even California voter's just reaffirmed) and many companies could stand to make a lot of money off of the cheap workforce. The prior Trump administration focused on breaking families up, but this time they promise to include everyone in the family, no matter the immigration status, even denaturalization of American citizens. How will it be payed for? The operation is projected to cost upwards of $300 billion or much more and this has frequently been cited as a major reason that the invisioned plans will falter. If Elon Musk's Department of Government Efficiency's (DOGE) plans for freeing up $2 trillion of the yearly federal budget succeeds, there appears to be plenty of money to reallocate. Even without these slush funds, Trump has already declared that the operation has "no price tag." Some recent and historical parallels In China, many Uyghurs have been put into forced labor prison camps for decades now, and the Xinjiang region has long been at the technological vanguard of state of the art large scale surveillance and population control research and development. American and Western tech companies can take what they have learned in Xinjiang and finally bring it home for wide spread use domestically. During the Holocaust, many "good Germans" helped to report their neighbors, and already Americans are arguing whether it is ethical to call in ICE on theirs. America has a long history of anti-immigrant campaigns and attacking ethnic communities, including Operation Wetback in 1953-54, the Japanese Internment Camps during WWII and tracing back to events like the Trail of Tears and traditional American slavery, to name a few. Hoping for the best While the years of preparation for Project 2025, combined with the GOP's complete control of all of the branches of the federal government suggests things are going to be harder to stop. It is good to remember that for all intents and purposes this will be an unimaginable kakistocracy and the revolving door of grifters is likely to spin even faster than it did during the first Trump administration. Major logistical questions remain - how fast can they actually scale up, for example, or deal with the massively inflated costs, bribes and outright theft chipping away at the budget. The chaos caused by replacing 50,000 federal employees with largely incompetent hacks while gutting government agencies left and right alone will likely slow down any plans significantly due to the complex and tightly coupled systems that make up the US government. Fortunately the ACLU and many other organizations and Democratic state governments vow to legally fight back against the upcoming operations. And lastly, it is likely that there will be large scale public action. Whether it is in the form of protests, strikes or imaginative work arounds for whatever crackdowns that are coming, there many Americans who will push back.

Minister in the Presidency Khumbudzo Ntshavheni had no sympathy for illegal miners trapped in a shaft in Stilfontein, saying it is not the government's job to retrieve the bodies of criminals, and law enforcement officers' lives should not be risked to help criminals.

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Wedding bells rang at Trinity Methodist Church yesterday afternoon (Nov 12) as couples renewed their vows — but it was all for a scene in local director Jack Neo's upcoming movie, I Want To Be Boss. Among them, however, was an actual couple who have been married for more than six decades — Jack's very own parents. "I want to leave a beautiful memory of them in the movie," the 64-year-old actor-director told local media on the set. His mum, 86, was dressed in a white wedding gown with tulle sleeves, while Jack's father, 91, looked sharp in a black suit with a matching bowtie. Jack added: "I especially requested that my mum put on a wedding dress today for this scene, because the first time they wore their wedding outfits, I only got to see them in photos. I can see them now."

After his success with Messiah in 1742 A.D. he never performed an Italian opera again. Handel was only partly successful with his performances of English Oratorio on mythical or biblical themes, but when he arranged a performance of Messiah to benefit the Foundling Hospital (1750) the critique ended. The pathos of Handel's oratorio is an ethical one, they are hallowed not by liturgical dignity but by the moral ideals of humanity. Almost blind, and having lived in England for almost fifty years, he died a respected and rich man. -- Handel is regarded as one of the greatest composers of all time, not only because of his Water Music, and Music for the Royal Fireworks. But since the late 1960s, with the revival of baroque music and original instrument interest in Handel's opera seria has revived too. Handel composed forty operas in about thirty years; some are considered as masterpieces, with many sweeping arias and much admired improvisations. His operas contain remarkable human characterization, by a composer not known for his love affairs. -- Messiah: (HWV 56) is an English-language oratorio composed in 1741 by George Frideric Handel, with a scriptural text compiled by Charles Jennens from the King James Bible, and from the Psalms included with the Book of Common Prayer (which are worded slightly differently than their King James counterparts). It was first performed in Dublin on 13 April 1742, and received its London premiere nearly a year later. After an initially modest public reception the oratorio gained in popularity, eventually becoming one of the best-known and most frequently performed choral works in Western music.

To download the entire Tyndale Bible click here. You will need the Adobe Acrobat Reader for this file. To read a particular book, click on the appropriate link below: -- About the Wesley Center Online -- The Wesley Center Online web site is a collection of historical and scholarly resources about the Wesleyan Tradition, theology, Christianity, and the Nazarene church. Copyright © 1993-2011. Wesley Center for Applied Theology, c/o Northwest Nazarene University. All Rights Reserved.

Relatively little is known of the life of Irenaeus. As a boy he had, as he delighted to point out, listened to the sermons of the great bishop and martyr, Polycarp of Smyrna, who was regarded as a disciple of the apostles themselves. Here he came to know, 'the genuine unadulterated gospel', to which he remained faithful throughout his life. Perhaps he also accompanied Polycarp on his journey to Rome in connection with the controversy over the date of celebrating Easter (154 CE). Later he went as a missionary to southern Gaul, where he became a presbyter at Lyons. A Catholic Encyclopedia article is online at St. Irenaeus. Irenaeus was absent from the city when the persecution there reached its zenith. It seems that he had been sent to Rome by the Gallican churches in order to confer with Pope Eleutherus, perhaps as a mediator in the Montanist disputes. Evidently Irenaeus stayed in Rome for just a short time, and soon after the end of the persecution we find him again in Lyons as the successor to Bishop Pothinus (178). When and how he died is unknown to us. Jerome and others state that he died as a martyr in the persecution under the Emperor Septimus Severus (202), but there is no certainty about this tradition. In short, we know Irenaeus almost solely from his writings, and these have not been preserved in their entirety. ... The era in which Irenaeus lived was a time of expansion and inner tensions in the church. In many cases Irenaeus acted as mediator between various contending factions. The churches of Asia Minor (where he was probably born) continued to celebrate Easter on the same date (the 14th of Nisan) as the Jews celebrated Passover, whereas the Roman Church maintained that Easter should always be celebrated on a Sunday (the day of the Resurrection). Mediating between the parties, Irenaeus stated that differences in external factors, such as dates of festivals, need not be so serious as to destroy church unity. Irenaeus adopted a totally negative and unresponsive attitude, however, toward Marcion, a schismatic leader in Rome, and toward the Valentinians, a fashionable intellectual Gnostic movement in the rapidly expanding church that espoused dualism. Because Gnosticism was overcome by the Orthodox Church, Gnostic writings were largely obliterated. In reconstructing Gnostic doctrines, therefore, modern scholars relied to a great extent on the writings of Irenaeus, who summarized the Gnostic views before attacking them. After the discovery of the Gnostic library near Nag Hammadi in Egypt in the 1940s (see Robinson), respect for Irenaeus increased. He was proved to have been extremely precise in his report of the doctrines he rejected. The oldest lists of bishops also were countermeasures against the Gnostics, who said that they possessed a secret oral tradition from Jesus himself. Against such statements Irenaeus maintains that the bishops in different cities are known as far back as the Apostles - and none of them was a Gnostic - and that the bishops provided the only safe guide to the interpretation of the Scriptures. With these lists of bishops the later doctrine of "the apostolic succession" of the bishops could be linked.

The Apostle Paul - The apostle Paul was one of the most famous citizens of the Roman Empire and without question one of the most influential individuals in history. He was used by the Lord in his missionary and evangelistic activities to set in motion a great deal of the organization known as the Christian Church, the Body of Christ on earth, to the extent that billions of human beings have been directly or indirectly affected by his ministry. Under the inspiration of the Holy Spirit, he wrote the foundation documents for the Christian way of life, the Word of God which has changed the lives of millions. -- Paul's Education: Paul was educated by his mother until the age of five. From age five to ten he studied with his father in the Hebrew scriptures and traditional writings. At the same time, being a Roman citizen and living in a Greek and Roman environment, he received a thorough education in the Greek language, history, and culture. He was sent to Jerusalem at about the age of ten to attend the rabbinical school of Gamaliel, who was the son of Simeon the son of Hillel. Gamaliel was a most eminent rabbi who was mentioned both in the Talmud and in the New Testament (Acts 5:24-40; 22:3). Gamaliel was called Rabban - one of only seven teachers so called. He was a Pharisee, but he rose above party prejudice. He composed a prayer against the Christian "heretics". He lived and died a Jew. At this time, Herod was dead, and the Romans had complete control of Judea, hence, there was Roman money, language, and culture. The Jews, therefore, were inclined to cling more closely to their religion as the center of unity. [Refer to the topic: Judean History] There were two great rabbinical schools, those of Hillel and Schammai. Hillel, the grandfather of Gamaliel, held that tradition was superior to the Law. The school of Schammai despised traditionalists, especially when there teachings clashed with the writings of Moses. The religious school of Gamaliel (Hillel) was chiefly oral and usually had a prejudice against any book but Scripture. They used a system of Scriptural exegesis, and Josephus in his writings expressed the wish to have such a power of exegesis. When school was in session, learned men met and discussed scriptures, gave various interpretations, suggested illustrations, and quoted precedents. The students were encouraged to question, doubt, even contradict. -- When Paul became a Christian, his very thorough education was enormously helpful. He was able to assimilate Christian doctrines rapidly and relate them accurately to the Scripture teaching he had received. From his education, both from Gamaliel and in the desert from the Lord Jesus Christ, Paul developed a divine viewpoint attitude toward human history. Paul knew that the existence of God can easily be perceived by anyone, that man can become aware of God, but that many men's deliberate sin halted this good beginning by immoral activities which accompanied their idolatry. Therefore, Paul had an intense hatred of idolatry of any kind. Paul's teaching shows that the only reality is God. Idolatry distorts man's conception of the world and external nature. Idolatry is the enemy of mankind. Paul knew the law of growth of human nature. As a Roman, Tarsian, Hebrew, and culturally Greek, he knew of the many distortions of the life of his society. As a nation becomes unhealthy, development is halted. Societies errors as to the nature of God and the true relation of God to man prevented nations from getting rid of their besetting evil. The books of Acts is the chief authoritative record for the ministries of Paul and the other apostles. For a brief outline of Paul's ministry, see the Chronological Table of Paul's Ministry. The most thorough, accurate, and interesting secular work on Paul is The Life and Epistles of St. Paul, by Conybeare and Howson.

Psalms 118:19-29 Open to me the gates of righteousness: I will go into them, and I will praise the LORD: This gate of the LORD, into which the righteous shall enter. I will praise thee: for thou hast heard me, and art become my Salvation. The stone which the builders refused is become the head stone of the corner. This is the LORD'S doing; it is marvellous in our eyes. This is the day which the LORD hath made; we will rejoice and be glad in it. Save now, I beseech thee, O LORD: O LORD, I beseech thee, send now prosperity [lit. wellbeing]. Blessed be He (Jesus) that cometh in the Name of the LORD: we have blessed you out of the House of the LORD. God is the LORD, which hath shewed us [resurrection - sunrise] light: bind the sacrifice with cords, even unto the horns of the altar. Thou art my God, and I will praise thee: thou art my God, I will exalt thee. O give thanks unto the LORD; for He is good: for His mercy endureth for ever. -- Holy Bible

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It is easy to see why text messaging is the most-used data service across the world. We send and receive five times as many texts compared with the number of phone calls each day, according to mobile intelligence firm Informate.

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The food and beverage industry is a tough game. Sixty percent of restaurants don’t make it past their first year, and 80 percent go out of business within five years. Those are hard odds.

Franchising takes some of the risks out of the equation by giving you a proven model to work with. But being a franchisor with a proven model under your arm does not mean you’re suddenly bulletproof or immune to the laws of economics. If you start making unforced errors, you are going to fail.

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“Young man, in mathematics you don’t understand things, you just get used to them.” —John von Neumann¹

This, in a sense, is at the heart of why mathematics is so hard. Math is all about abstraction, about generalizing the stuff you can get a sense of to apply to crazy situations about which you otherwise have no insight whatsoever. Take, for example, one way of understanding the manifold structure on SO(3), the special orthogonal group on 3-space. In order to explain what I’m talking about, I’ll have to give several definitions and explanations and each, to a greater or lesser extent, illustrates both my point about abstraction and von Neumann’s point about getting used to things.

First off, SO(3) has a purely algebraic definition as the set of all real (that is to say, the entries are real numbers) 3 × 3 matrices A with the property A^TA = I and the determinant of A is 1. That is, if you take A and flip rows and columns, you get the transpose of A, denoted A^T; if you then multiply this transpose by A, you get the identity matrix I. The determinant has its own complicated algebraic definition (the unique alternating, multilinear functional…), but it’s easy to compute for small matrices and can be intuitively understood as a measure of how much the matrix “stretches” vectors. Now, as with all algebraic definitions, this is a bit abstruse; also, as is unfortunately all too common in mathematics, I’ve presented all the material slightly backwards.

This is natural, because it seems obvious that the first thing to do in any explication is to define what you’re talking about, but, in reality, the best thing to do in almost every case is to first explain what the things you’re talking about (in this case, special orthogonal matrices) really are and why we should care about them, and only then give the technical definition. In this case, special orthogonal matrices are “really” the set of all rotations of plain ol’ 3 dimensional space that leave the origin fixed (another way to think of this is as the set of linear transformations that preserve length and orientation; if I apply a special orthogonal transformation to you, you’ll still be the same height and width and you won’t have been flipped into a “mirror image”). Obviously, this is a handy thing to have a grasp on and this is why we care about special orthogonal matrices. In order to deal with such things rigorously it’s important to have the algebraic definition, but as far as understanding goes, you need to have the picture of rotations of 3 space in your head.

Okay, so I’ve explained part of the sentence in the first paragraph where I started throwing around arcane terminology, but there’s a bit more to clear up; specifically, what the hell is a “manifold”, anyway? Well, in this case I’m talking about differentiable (as opposed to topological) manifolds, but I don’t imagine that explanation helps. In order to understand what a manifold is, it’s very important to have the right picture in your head, because the technical definition is about ten times worse than the special orthogonal definition, but the basic idea is probably even simpler. The intuitive picture is that of a smooth surface. For example, the surface of a sphere is a nice 2-dimensional manifold. So is the surface of a donut, or a saddle, or an idealized version of the rolling hills of your favorite pastoral scene. Slightly more abstractly, think of a rubber sheet stretched and twisted into any configuration you like so long as there are no holes, tears, creases, black holes or sharp corners.

In order to rigorize this idea, the important thing to notice about all these surfaces is that, if you’re a small enough ant living on one of these surfaces, it looks indistinguishable from a flat plane. This is something we can all immediately understand, given that we live on an oblate spheroid that, because it’s so much bigger than we are, looks flat to us. In fact, this is very nearly the precise definition of a manifold, which basically says that a manifold is a topological space (read: set of points with some important, but largely technical, properties) where, at any point in the space, there is some neighborhood that looks identical to “flat” euclidean space; a 2-dimensional manifold is one that looks locally like a plane, a 3-dimensional manifold is one that looks locally like normal 3-dimensional space, a 4-dimensional manifold is one that looks locally like normal 4-dimensional space, and so on.

In fact, these spaces look so much like normal space that we can do calculus on them, which is why the subject concerned with manifolds is called “differential geometry”. Again, the reason why we would want to do calculus on spaces that look a lot like normal space but aren’t is obvious: if we live on a sphere (as we basically do), we’d like to be able to figure out how to, e.g., minimize our distance travelled (and, thereby, fuel consumed and time spent in transit) when flying from Denver to London, which is the sort of thing for which calculus is an excellent tool that gives good answers; unfortunately, since the Earth isn’t flat, we can’t use regular old freshman calculus.² As it turns out, there are all kinds of applications of this stuff, from relatively simple engineering to theoretical physics.

So, anyway, the point is that manifolds look, at least locally, like plain vanilla euclidean space. Of course, even the notion of “plain vanilla euclidean space” is an abstraction beyond what we can really visualize for dimensions higher than three, but this is exactly the sort of thing von Neumann was talking about: you can’t really visualize 10 dimensional space, but you “know” that it looks pretty much like regular 3 dimensional space with 7 more axes thrown in at, to quote Douglas Adams, “right angles to reality”.

Okay, so the claim is that SO(3), our set of special orthogonal matrices, is a 3-dimensional manifold. On the face of it, it might be surprising that the set of rotations of three space should itself look anything like three space. On the other hand, this sort of makes sense: consider a single vector (say of unit length, though it doesn’t really matter) based at the origin and then apply every possible rotation to it. This will give us a set of vectors based at the origin, all of length 1 and pointing any which way you please. In fact, if you look just at the heads of all the vectors, you’re just talking about a sphere of radius 1 centered at the origin. So, in a sense, the special orthognal matrices look like a sphere. This is both right and wrong; the special orthogonal matrices do look a lot like a sphere, but like a 3-sphere (that is, a sphere living in four dimensions), not a 2-sphere (i.e., what we usually call a “sphere”).

In fact, locally SO(3) looks almost exactly like a 3-sphere; globally, however, it’s a different story. In fact, SO(3) looks globally like , which requires one more excursion into the realm of abstraction. , or real projective 3-space, is an abstract space where we’ve taken regular 3-space and added a “plane at infinity”. This sounds slightly wacky, but it’s a generalization of what’s called the projective plane, which is basically the same thing but in a lower dimension. To get the projective plane, we add a “line at infinity” rather than a plane, and the space has this funny property that if you walk through the line at infinity, you get flipped into your mirror image; if you were right-handed, you come out the other side left-handed (and on the “other end” of the plane). But not to worry, if you walk across the infinity line again, you get flipped back to normal.

Okay, sounds interesting, but how do we visualize such a thing? Well, the “line at infinity” thing is good, but infinity is pretty hard to visualize, too. Instead we think about twisting the sphere in a funny way:

You can construct the projective plane as follows: take a sphere. Imagine taking a point on the sphere, and its antipodal point, and pulling them together to meet somewhere inside the sphere. Now do it with another pair of points, but make sure they meet somewhere else. Do this with every single point on the sphere, each point and its antipodal point meeting each other but meeting no other points. It’s a weird, collapsed sphere that can’t properly live in three dimensions, but I imagine it as looking a bit like a seashell, all curled up on itself. And pink.

This gives you the real projective plane, . If you do the same thing, but with a 3-sphere (again, remember that this is the sphere living in four dimensions), you get . Of course, you can’t even really visualize or, for that matter, a 3-sphere, so really visualizing is going to be out of the question, but we have a pretty good idea, at least by analogy, of what it is. This is, as von Neumann indicates, one of those things you “just get used to”.

Now, as it turns out, if you do the math, SO(3) and look the same in a very precise sense (specifically, they’re diffeomorphic). On the face of it, of course, this is patently absurd, but if you have the right picture in mind, this is the sort of thing you might have guessed. The basic idea behind the proof linked above is that we can visualize 3-space as living inside 4-space (where it makes sense to talk about multiplication); here, a rotation (remember, that’s all the special orthogonal matrices/transformations really are) is just like conjugating by a point on the sphere. And certainly conjugating by a point is the same as conjugating by its antipodal point, since the minus signs will cancel eachother in the latter case. But this is exactly how we visualized , as the points on the sphere with antipodal points identified!

I’m guessing that most of the above doesn’t make a whole lot of sense, but I would urge you to heed von Neumann’s advice: don’t necessarily try to “understand” it so much as just to “get used to it”; the understanding can only come after you’ve gotten used to the concepts and, most importantly, the pictures. Which was really, I suspect, von Neumann’s point, anyway: of course we can understand things in mathematics, but we can only understand them after we suspend our disbelief and allow ourselves to get used to them. And, of course, make good pictures.

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¹ This, by the way, is my second-favorite math quote of the year, behind my complex analysis professor’s imprecation, right before discussing poles vs. essential singularities, to “distinguish problems that are real but not serious from those that are really serious.”

² As a side note, calculus itself is a prime example of mathematical abstraction. The problem with the world is that most of the stuff in it isn’t straight. If it were, we could have basically stopped after the Greeks figured out a fair amount of geometry. And, even worse, not only is non-straight stuff (like, for example, a graph of the position of a falling rock plotted against time) all over the place, but it’s hard to get a handle on. So, instead of just giving up and going home, we approximate the curvy stuff in the world with straight lines, which we have a good grasp of. As long as we’re dealing with stuff that’s curvy (rather than, say, broken into pieces) this actually works out pretty well and, once you get used to it all, it’s easy to forget what the whole point was, anyway (this, I suspect, is the main reason calculus instruction is so uniformly bad; approximating curvy stuff with straight lines works so well that those who who are supposed to teach the process lose sight of what’s really going on).