Day 2: From Birds to Machines... and into the Future
On Tuesday morning, Cluster 11’s Prof. Ricardo Sanfelice presented his lecture, “From birds to machines…and into the future,” which covered intelligent transportation and the discoveries leading up to the machines we have today. First, Prof. Sanfelice elaborated on the inspiring creatures on which inventors based the first flying machine, birds! Birds can be considered a natural flying machine. Since they are descendants of dinosaurs, having been on the earth for a while, inventors observed their motions as they soar through the sky. They discovered two main flying modes: soaring and flapping. After watching a video on the anatomy of the flight, the audience better understood the motion of the muscles of the birds’ wings required to keep them in the air for long amounts of time. Birds interestingly have specific flight feathers that are actually asymmetrical to give more surface area on the part pushing against the air. I find it captivating that nature adjusts itself to improve the situation.
After learning about the inspiration behind machines, we were shown the many machine attempts ranging as early as 350 BC! Some examples include the ‘Flying Pigeon’, a self propelled bird-shaped model, and the ‘Sky Lantern’, where the lamp heats the air for propulsion.
Although, with time, inventors realized their machines needed control and wanted to create a more precise path and be able to rotate their directions. The hang glider system came about equipped with Flying gliders, and so did hydrogen hot air balloons. The room filled with laughter as Prof. Sanfelice showed us a compilation video of the mostly failed attempts. Although the initial purpose of these machines were mostly made for war equipment, they have many other uses today such as traveling. They paved the road of discoveries that lead us to the machines we have today! It took trial and error, and a lot of patience to progress our knowledge in machine engineering.
Even though we have come so far from the first machine attempts like the ‘Flying Pigeon’, there are still so many possibilities to create for the future. Prof. Sanfelice introduced the idea of developing automations and listed some ideas that companies are developing, such as an airplane-helicopter taxi system, creating air machines that travel faster than sound, and SpaceX. I’m sure most of us will now be keeping up with the new and improved technologies in the near future. And as always, I am looking forward to our next discovery lecture, excited for whatever new topic I can know more about.
- Hannah Chang
After learning about the inspiration behind machines, we were shown the many machine attempts ranging as early as 350 BC! Some examples include the ‘Flying Pigeon’, a self propelled bird-shaped model, and the ‘Sky Lantern’, where the lamp heats the air for propulsion.
Although, with time, inventors realized their machines needed control and wanted to create a more precise path and be able to rotate their directions. The hang glider system came about equipped with Flying gliders, and so did hydrogen hot air balloons. The room filled with laughter as Prof. Sanfelice showed us a compilation video of the mostly failed attempts. Although the initial purpose of these machines were mostly made for war equipment, they have many other uses today such as traveling. They paved the road of discoveries that lead us to the machines we have today! It took trial and error, and a lot of patience to progress our knowledge in machine engineering.
Even though we have come so far from the first machine attempts like the ‘Flying Pigeon’, there are still so many possibilities to create for the future. Prof. Sanfelice introduced the idea of developing automations and listed some ideas that companies are developing, such as an airplane-helicopter taxi system, creating air machines that travel faster than sound, and SpaceX. I’m sure most of us will now be keeping up with the new and improved technologies in the near future. And as always, I am looking forward to our next discovery lecture, excited for whatever new topic I can know more about.
- Hannah Chang
Day 4: Infinity: Decimals, Lengths and Size
After all 12 clusters filed into the packed Discovery Lecture hall, Professor Shaowei Chen kicked off the day with a quick roll call, including the first (and only) Cluster 1 chant, surprising Prof. Chen enough that he offered to record the chant next time. However, no such recording ever happened (since this was the last chant Cluster 1 would do), and soon Professor Bob Hingtgen’s lecture began with a Spiderman meme regarding infinity, aleph null, and omega — all topics yet to be explored.
Firstly, Bob addressed the room with a question: is a line made up of infinitely many points? Bob elaborated on this, talking about how any line could be divided up into infinitely many, small, subintervals, while simultaneously giving calculus students terrifying flashbacks of computing limits. So, does infinity (one spiderman down!) times zero equal one? Just like many other math questions, this one, too, had no decisive answer as the result varied from finite to indeterminate based on the mathematical context.
Next, we learned about adding two finite intervals, and the differences between adding open and closed intervals. Although at the time this topic seemed relatively easy to understand, it was just a sign of things to come. Bob then started talking about infinitely repeating decimals like 0.999… and 1.000… by breaking up each place value of the decimals and treating each one like an interval. This intuitively made it clear that numbers like the aforementioned repeating decimals were actually the same number — their only difference was that they were reaching their true “address,” or value, from different directions (the left and the right.) But, by convention, mathematicians tend to use the “left representation” of numbers rather than the right. Jumping off of repeating decimals, Bob gave us the formula for evaluating infinite geometric series.
Then, Bob transitioned to a different topic — counting. Using a world-class artistic depiction of the elusive hover sheep, he explained how counting always involves forming a pairing, or bijection, between the objects that you are trying to count and the natural numbers (1, 2, 3, …).
Bob also explained the cardinality of a set or the number of elements that it has. For sets with finitely many elements, this may be easy to count, but for infinite sets like the rational numbers, it gets more complex. In fact, the number of elements in the set of natural numbers is denoted by aleph null, (two spidermen down!) which is a type of countable infinity. Going back to the previous corollary about bijections, Bob showed us how our typical understanding of bijections does not apply to infinite sets by forming a bijection between the natural numbers and the positive even numbers, destroying the dreams of Cluster 12 in the process. As the natural numbers “obviously” have more elements than the positive even numbers, but a bijection can still be formed between the sets, this implies that the corollary is false for infinite sets.
Extending this experiment, Bob talked about Hilbert’s Hotel, a thought experiment about a hotel with infinite rooms, all of which are full. Bob showed how if some countable infinity (like the natural numbers) of people arrived at the hotel, space could still be made at the hotel for all the new guests. In fact, if a countable infinity of buses, each holding a countable infinity of people (actually, this setup models the rational numbers) shows up, a room can still be made for all new guests. Extending this result, it can be shown that if a countable infinity of a countable infinity… of a countable infinity of similar buses shows up, the hotel can still accommodate everyone.
This is a very surprising result — how can (aleph null)^(aleph null) people show up at the hotel, but still be accommodated in just aleph null rooms? Does this imply that aleph null = (aleph null)^(aleph null)? Actually, Bob showed us a proof that the former might even be bigger than the latter, using a mapping involving prime factorization!
Continuing with Hilbert’s Hotel, Bob explained that if a single bus shows up with infinite people, each named by an infinitely long string of 0’s and 1’s (representing the real numbers), there would be no way to fit everyone into the hotel because of Cantor’s Diagonalization Argument. In fact, the size of the real numbers is so big, that it is referred to as an uncountable infinity.
Finally, Bob briefly explained the Cantor Set, which, by construction, seems to have the same size as the real numbers, but also no size at the same time. Looping back to the beginning of his presentation, Bob talked about adding the intervals of the Cantor Set to themselves and ended up showing that 0 + 0 = 2, truly demonstrating how even fundamental ideas like interval addition can break down in the face of infinity.
Ending his lecture to a thundering round of applause, Bob answered a few questions and wrapped up my favorite discovery lecture so far (though I may be biased).
- Advaith Mopuri
Firstly, Bob addressed the room with a question: is a line made up of infinitely many points? Bob elaborated on this, talking about how any line could be divided up into infinitely many, small, subintervals, while simultaneously giving calculus students terrifying flashbacks of computing limits. So, does infinity (one spiderman down!) times zero equal one? Just like many other math questions, this one, too, had no decisive answer as the result varied from finite to indeterminate based on the mathematical context.
Next, we learned about adding two finite intervals, and the differences between adding open and closed intervals. Although at the time this topic seemed relatively easy to understand, it was just a sign of things to come. Bob then started talking about infinitely repeating decimals like 0.999… and 1.000… by breaking up each place value of the decimals and treating each one like an interval. This intuitively made it clear that numbers like the aforementioned repeating decimals were actually the same number — their only difference was that they were reaching their true “address,” or value, from different directions (the left and the right.) But, by convention, mathematicians tend to use the “left representation” of numbers rather than the right. Jumping off of repeating decimals, Bob gave us the formula for evaluating infinite geometric series.
Then, Bob transitioned to a different topic — counting. Using a world-class artistic depiction of the elusive hover sheep, he explained how counting always involves forming a pairing, or bijection, between the objects that you are trying to count and the natural numbers (1, 2, 3, …).
Bob also explained the cardinality of a set or the number of elements that it has. For sets with finitely many elements, this may be easy to count, but for infinite sets like the rational numbers, it gets more complex. In fact, the number of elements in the set of natural numbers is denoted by aleph null, (two spidermen down!) which is a type of countable infinity. Going back to the previous corollary about bijections, Bob showed us how our typical understanding of bijections does not apply to infinite sets by forming a bijection between the natural numbers and the positive even numbers, destroying the dreams of Cluster 12 in the process. As the natural numbers “obviously” have more elements than the positive even numbers, but a bijection can still be formed between the sets, this implies that the corollary is false for infinite sets.
Extending this experiment, Bob talked about Hilbert’s Hotel, a thought experiment about a hotel with infinite rooms, all of which are full. Bob showed how if some countable infinity (like the natural numbers) of people arrived at the hotel, space could still be made at the hotel for all the new guests. In fact, if a countable infinity of buses, each holding a countable infinity of people (actually, this setup models the rational numbers) shows up, a room can still be made for all new guests. Extending this result, it can be shown that if a countable infinity of a countable infinity… of a countable infinity of similar buses shows up, the hotel can still accommodate everyone.
This is a very surprising result — how can (aleph null)^(aleph null) people show up at the hotel, but still be accommodated in just aleph null rooms? Does this imply that aleph null = (aleph null)^(aleph null)? Actually, Bob showed us a proof that the former might even be bigger than the latter, using a mapping involving prime factorization!
Continuing with Hilbert’s Hotel, Bob explained that if a single bus shows up with infinite people, each named by an infinitely long string of 0’s and 1’s (representing the real numbers), there would be no way to fit everyone into the hotel because of Cantor’s Diagonalization Argument. In fact, the size of the real numbers is so big, that it is referred to as an uncountable infinity.
Finally, Bob briefly explained the Cantor Set, which, by construction, seems to have the same size as the real numbers, but also no size at the same time. Looping back to the beginning of his presentation, Bob talked about adding the intervals of the Cantor Set to themselves and ended up showing that 0 + 0 = 2, truly demonstrating how even fundamental ideas like interval addition can break down in the face of infinity.
Ending his lecture to a thundering round of applause, Bob answered a few questions and wrapped up my favorite discovery lecture so far (though I may be biased).
- Advaith Mopuri