beila
2025
1유향곡면이 아닌 곡면의 예시는 뫼비우스 띠와 클라인의 병이 있다. 지난 글에서는 그래프와 그 대칭군에 대해 알아보았다. 이번 글에서는, 이 시리즈의 또 다른 주인공인 곡면과 그 대칭군에 대해 알아보려고 한다. 2차원 수학적 대상인 곡면은, 김현규 교수님의 글 에서처럼 수영장에서 볼 수 있는 비치볼과 튜브들로 생각해 볼 수 있다. 그림1은 에서 먼저 제시되었다. 여기서 비치볼을 “\(0\) 인용 튜브”라고 생각하면, \(0\) 이상의 정수 \(n\)에 대해, \(n\)인용 튜브는 안팎을 구분할 수 있는 소위…
I recently visited Puerto Rico. As I dove into its geography and history, I fell in a rabbit hole to understand why the Caribbean is the way it is today. It’s fascinating. So please get into this adventure with me! Here’s what you’re going to discover: Why Mexico’s economy is the way it is. Why the…
I also finally got around to finish editing my interview with Eva Maydell, the Member of the European Parliament. Here it is! Podcasts take me a lot of work, so I still need to figure out how to make them in a way that makes sense for you and me. Keep reading with a 7-day free trial Subscribe to…
It’s a miracle: The $10-billion James Webb telescope was successfully launched, deployed, and activated. It’s now sending unbelievable pictures that you might have already seen: The images are not just beautiful. They’re revealing the secrets of the universe. They’re charting uncharted territories.…
Geography and history shouldn’t be memorized. They should be understood. I recall vividly the middle school class about Vikings. In French history and geography class, we had learned about the fall of Rome (But why did it fall?), the arrival of the Franks from the east (But why did they arrive?),…
Europe is geographically much closer to Africa than America. Northern Africa has been part of the Eurasian culture since Alexander the Great conquered Egypt, over two millennia ago. Uncharted Territories The Dictatorship of the Nile I’m trying something new this week. I’m publishing the free and…
A few weeks ago, in Geography Is the Chessboard of History, we discovered a few ways Geography determines a lot about History. Since then, I’ve done a deep dive on Switzerland, and on Spain, France, and Germany. Today, we’re going global: we’ll first have a look at the US, and from there we’ll…
By the time you finish reading this article, you will see from a different light news like the ransomware attack on Colonial, the growth of Bitcoin, the rules of the creator economy, the global negotiations on tax rates, or even this. I hope you enjoy reading it as much as I enjoyed writing it!…
We could have instant drone delivery. We could have flying cars. We could have energy too cheap to meter. We could forget about diseases. We don’t because we’re too scared. Ankylosaurus Inc Svieta stops. The lights turn back on. She turns around to face the room. They’re looking at each other around…
The elders reminisce: Our great-great-grandparents once worked together, but they had a falling out. Nobody remembers when, or why. But every Martel knows you can’t argue with Hasfiris. Hasfiris know better than to reason with a filthy Martel. The Martels know this because they live by the hammer.…
1. Foot-Shooters Boris Johnson is out. This is the best leader the UK could muster after world-class characters like Theresa May and David Cameron, father of the biggest shot in the foot the UK has triggered in a long time: Brexit. In this critical time of economic, geopolitical, and pandemic…
1. Bolivarians Some Latin-American leaders have a habit of criticizing Spain for its invasions centuries ago. Maybe inspired by that, some Latin-American friends teased me in grad school: ME: Where are you from? LATAM FRIEND: I’m from [Former Spanish colony], and you? ME: Spain and France. LAF: Ah!…
In Should Everybody Learn English?, I make the case that English is unstoppable. It’s only a matter of time before it becomes the one language that everybody speaks. I’ve been looking for data to confirm or deny this hypothesis since. I suffer from confirmation bias, so I must be blinded to signs…
The Free World fears China’s rise. It will soon be the largest economy in the world. People witness the speed of Chinese construction with awe. Balaji Srinivasan @balajis Or the construction of a train station in 9 hours, literally overnight: youtube.com1,500 Chinese workers build train station in…
You might have heard about the Urban-Rural Political Divide: In the US, people in cities vote more liberal, and in rural areas they vote more conservative. In fact, this rule is much more fundamental, and its ramifications might drive the political future of the world. Source: Karim Douieb via…
I write a lot about exchanging ideas: In last week’s premium article, I explored disagreement over the Internet. I have a series on how to give and get feedback. I’ve written other articles on these topics. Today, I talk about why debate is broken online and propose a way to improve it. Why do I pay…
This is the second premium article of this week. Keep reading with a 7-day free trial Subscribe to Uncharted Territories to keep reading this post and get 7 days of free access to the full post archives. Already a paid subscriber? Sign in
In The Internet and Blockchain Will Kill Nation-States and The End of Nation-States, I explain how communication technologies created the nation-state, and why the Internet will replace it with something else. But what will emerge in its wake? This month, I will continue this series of articles.…
Earlier this week, we saw why decentralization shows the way for the future of democracy. But what decentralized information mechanisms can we create to improve democracy? One hint comes from the history of finance. 1. Sumerian Mints Your neighbor is knocking on the door: M: Hey Camile! What’s up?…
Remote work is one of the most hotly debated topics in the corporate world. Will people go back to the office? Stay home? The single biggest argument in favor of the office is the watercooler: The idea that employees can randomly meet and strike up conversations that end in collaborations. A crucial…
Education today doesn’t look that different from education decades, or even centuries ago1. Sure, you have screens now, sometimes. And remote education has also changed everything—frequently for the worse. But children2 are still sitting 8 hours a day, five days a week, for years on end in front of…
Sorry for the confusion earlier this week. I mistakenly sent the free article to premium subscribers only, and had to send it again. One additional note: I’ve never mentioned a time off policy. My goal is to take four weeks off a year from the newsletter—either for true time off or to catch up on…
Digital images of apes have sold at over $3M a piece. Famous people rushed to get one. Buyers of Bored Apes from BAYC, Bored Ape Yacht Club. Source A pixelated jpeg sold for over $20M. Source 20-second NBA highlights sold for up to $230k. The image below, Everydays: the First 5000 Days, sold for…
Of these two companies, which one would you invest in? In which one would you want to work? In last week’s free article, I explained why network effects companies like aggregators and platforms account for 70% of the value created in tech. Network effects make your revenue look like the company on…
This is the follow-up to yesterday’s article, Generative AI: Everything You Need to Know. As you can tell, some industries are about to disappear. Marketing copy is about to disappear. Stock image companies are about to disappear. Corporate photoshoot photographers are about to disappear. And…
Java has a larger population than Russia. 151M people in Java vs 143M in Russia And this is not just a story of “there are plenty of people in Asia”. Java has a bigger population than Japan! Even by Asian standards, Java is just extremely densely populated. Source: Alasdair Rae from this great…
Have you seen the videos? South Asia Index @SouthAsiaIndex Pakistan's devastating floods: - 1350 people killed - 50M people displaced - 900K livestock deaths - 1M houses washed away - 40+ reservoirs breached - 220+ bridges collapsed - 90% cropped damaged - $10B loss to economy - 1/3 country…
Nearly 600 million people speak Spanish and nearly 300 million speak Portuguese1. We know why: Portugal and Spain colonized America, making it Latin America, where most of these 900 million people live. About 60 million live in Europe2 and 70 million in Africa3. But why was it Portugal and Spain…
The Iberian Peninsula is a clear geographic unit, which would lead you to think it should be one country1. But as we saw in previous articles, mountains had a different idea. It turns out that most of the peninsula has coalesced around one single country, Spain. It has had separatist movements for a…
Last week we covered the GeoHistory of Spain. But as you could tell, it’s completely intertwined with that of Portugal. You can’t cover one without the other. Their histories are the same up to the 1200s, so you can read about what happened before in A Brief History of Spain. But Portugal has its…
Why did Spain, and not another country, conquer most of America? Why have so many Spanish regions pushed for independence in contemporary Spain, like Catalonia or the Basque Country? Why are Spain and Portugal not united? In Spain, what is the legacy of Muslims, who ruled the region for 800 years?…
All religions are equally sublime to the ignorant, useful to the politician, and ridiculous to the philosopher.—Lucretius1 If you travel to the southwest of France, to an area called Occitanie, and talk with a local about their region, they will tell you with defiance—and what Parisians would call a…
This is the 2nd article in the series about France: If you missed the 1st one, go read France’s Bowels: A Secret Medieval Sect, an Angry Pope, an Opportunistic Lord, a Genocide, and Europe’s Destiny. France is weird. Why is it the biggest sea country in the world? Why was it the most powerful? Why…
Victor Hugo addressing the Peace Congress of Paris on 21 August 1849 “A day will come when there will be no battlefields, but markets opening to commerce and minds opening to ideas. A day will come when the bullets and bombs are replaced by votes, by universal suffrage, by the venerable arbitration…
“There are two kinds of European nations. There are small nations and there are countries that have not yet realized they are small nations.”—Danish Finance Minister Kristian Jensen, talking about the UK in the conference Road To Brexit, 2017 The surface of the entire EU is about the size of the US,…
While hundreds of thousands of Ethiopians starve due to a government blockade, rebels are 250km away from the capital Addis Ababa. Last time those same rebels got there, they ousted the government and remained in power for 30 years. At the same time, Ethiopia and Egypt have been on the brink of war…
I’m trying something new this week. I’m publishing the free and premium articles together as one. The free version is the first part. The positive side is that you get the entire article about Egypt at once. The negative is that you get the entire article at once. Let me know if you prefer this or…
Given two values n1 and n2 where n1 < n2 and a root pointer to a Binary Search Tree. The task is to find all the keys of the tree in the range n1 to n2 in increasing order. Examples: Input: n1 = 10 , n2 = 22 Output: 12, 20 and 22. Explanation: The keys are 4, 8, 12, 20, and 22, So keys in range 10…
Given an integer array arr[] of size n, find the inversion count in the array. Two array elements arr[i] and arr[j] form an inversion if arr[i] > arr[j] and i < j. Note: The Inversion Count for an array indicates how far (or close) the array is from being sorted. If the array is already sorted, then…
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