by Jonathan Kujawa

Nearly four years ago here at 3QD we talked about how Euler became interested in a trifling little puzzle in 1736. He asked if it was possible to take a walk through the city of Königsberg, Prussia and cross each of its seven bridges once and only once. It seems like a silly thing to spend your time on and is obviously of no use to anyone. My former senator would have an aneurysm at the idea of funding such research. Fortunately for us, Frederick the Great was more of a visionary.
Being no slouch, not only did Euler solve the problem (spoiler: it’s not possible), he invented two major fields of modern mathematics in the process. Four years ago we followed one of those rabbit holes and were led to the field now called topology. It is the study of geometry when you don’t care about angles, distance, and the like. Topology is a rich and exciting area of modern research with all sorts of applications. It leads to subway maps, the use of homology to study the structure of huge data sets, motion planning for robots, and gives us that one easy trick for solving labyrinths like a Theseus.
The second rabbit hole is the one we follow today. It leads to another area of modern mathematics now called graph theory.
In mathematics, a graph is nothing more than a collection of nodes and connections between them (sometimes called vertices and edges, respectively). Graphs can be drawn where each node is a dot and each connection is a line connecting the corresponding dots. Read more »

To follow the popular discourse about the gender wage gap in the United States is to confront perpetual confusion. It is a confusion created at least in part by pronouncements of the type many of us have heard: “Women are paid only 82 cents for every dollar men earn! It is high time for women to earn equal pay for equal work!” Two sentences, each true standing alone, but in juxtaposition creating the impression that the
On occasions, while meandering the various English countryside and woodland paths, I have been pleasantly surprised to come across anglers. I have met fishermen dangling their lines in either a pond in some remote corner of the low-lying areas, or wading in water and casting a line down through the waters of a gently flowing river.



Unfortunately, you have a brain tumor. You don’t know it yet. Your doctor doesn’t know it yet. But you are beginning to have symptoms. The tumor is pressing on surrounding brain tissue and causing you develop a number of delusional beliefs. You believe you are the best swimmer in the world. You believe that dogs and cats are aliens. You believe that you invented the apostrophe. You also, as it happens, believe that you have a brain tumor.
“Taxi to Bethlehem, taxi to Jericho!” the man at a tourism kiosk is shouting, as I make my way from the tram to Jaffa Gate, known also as Hebron Gate, to Muslims as “Bab al Khalil,” or “door of the friend,” named after Hebron where the prophet Ibrahim/Abraham (Khalil al Allah “God’s Friend”) is laid to rest. Of significance too, is the association of this gate with King David’s (prophet Dawud’s) chamber, for followers of the three Abrahamic faiths: the crusaders named it “King David’s Gate.” It is one of the seven main stone portals of the walled city of Jerusalem.
Calls for a Manhattan Project–style crash effort to develop artificial intelligence (AI) technology are thick on the ground these days. Oren Etzioni, the CEO of the Allen Institute for Artificial Intelligence, recently issued such a call on 

Fans are the people who know the quotes, the dates of publication, the batting averages, the bassist on this album, the team that general manager coached before. I am not a fan. Don’t get me wrong. I’m full of enthusiasms. But I can’t match you statistic for statistic. I haven’t read the major author’s minor novel. I don’t care who the bassist was. You win. I’m an amateur.
When I watched the 2019 documentary on Apollo 11, it carried me back not to the summer of 1969, when it happened, but to the mid-1980s, when I was an undergrad. I was eight when Apollo 11 launched; of course I was aware of the space program and the moon landings, but I don’t have any memories of everyone gathering around to watch those first steps on another world. My parents weren’t particularly interested, and I don’t remember being caught by the spirit of the times myself.
cinematic representations of Muslims. Stage One features stereotyped figures (the taxi driver, terrorist, cornershop owner, or oppressed woman). Stage Two involves a portrayal that subverts and challenges those stereotypes. Finally, Stage Three is ‘the Promised Land, where you play a character whose story is not intrinsically linked to his race’. Does 
Our expectations sculpt neural activity, causing our brains to represent the outcomes of our actions as we expect them to unfold. This is consistent with a growing psychological literature suggesting that our experience of our actions is biased towards what we expect. —