Blending Math With Beauty
Question: What was the discovery process behind the Mandelbrot set?
Benoit Mandelbrot: The Mandelbrot set in a certain sense is a **** of a dream I had and an uncle of mine had since I was about 20. I was a student of mathematics, but not happy with mathematics that I was taught in France. Therefore, looking for other topics, an uncle of mine, who was a very well-known pure mathematician, wanted me to study a certain theory which was then many years old, 30 years old or something, but had in a way stopped developing. When he was young he had tried to get this theory out of a rut and he didn’t succeed, nobody succeeded. So, there was a case of two men, Julia, a teacher of mine, and Fatu, who had died, had a very good year in 1910 and then nothing was happening. My uncle was telling me, if you look at that, if you find something new, it would be a wonderful thing because I couldn’t – nobody could.
I looked at it and found it too difficult. I just could see nothing I could do. Then over the years, I put that a bit in the back of my mind until one day I read an obituary. It is an interesting story that I was motivated by an obituary, an obituary of a great man named Poincaré, and in that obituary this question was raised again. At that time, I had a computer and I had become quite an expert in the use of the computer for mathematics, for physics, and for many sciences. So, I decided, perhaps the time has come to please my uncle; 35 years later, or something. To please my uncle and do what my uncle had been pushing me to do so strongly.
But I approached this topic in a very different fashion than my uncle. My uncle was trying to think of something, a new idea, a new problem, a new way of developing the theory of Fatu and Julia. I did something else. I went to the computer and tried to experiment. I introduced a very high level of experiment in very pure mathematics. I was at IBM, I had the run of computers which then were called very big and powerful, but in fact were less powerful than a handheld machine today. But I had them and I could make the experiments. The conditions were very, very difficult, but I knew how to look at pictures. In fact, the reason I did not go into pure mathematics earlier was that I was dominated by visual. I tried to combine the visual beauty and the mathematics.
So, I looked at the picture for a long time in a very unsystematic fashion just to become acquainted in a kind of physical fashion with those extraordinary difficult and complicated shapes. Two were extraordinarily difficult. Computer graphics did not exist back then, but to have a machine which was – made it seem doable. And I started finding extraordinary complications, extraordinary structure, extraordinary beauty of both a theoretical kind, mathematical, and a visual kind. And collected observations of my trip in this new territory. When I presented that work to my colleagues, it was an explosion of interest. Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas. That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract. Pictures were completely eliminated from mathematics; in particular when I was young this happened in a very strong fashion.
Some mathematicians didn’t even perceive of the possibility of a picture being helpful. To the contrary, I went into an orgy of looking at pictures by the hundreds; the machines became a little bit better. We had friends who improved them, who wrote better software to help me, which was wonderful. That was the good thing about being at IBM. And I had this collection of observations, which I gave to my friends in mathematics for their pleasure and for simulation. The extraordinary fact is that the first idea I had which motivated me, that worked, is conjecture, a mathematical idea which may or may not be true. And that idea is still unproven. It is the foundation, what started me and what everybody failed to **** prove has so far defeated the greatest efforts by experts to be proven. In a certain sense it’s a very, very strange because the object itself is understandable, even for a child. If the object can be drawn by a child with new computers, with new graphic devices, and still the basic idea has not been proven.
But the development of it has been extraordinary, then it was slowed down a bit, and now again it is going up. New people are coming in and they prove extraordinary results which nobody was hoping to prove, and I am astonished and of course, very pleased by this development.
Question: What is the unproven conjecture that drove you?
Benoit Mandelbrot: The conjecture itself consists in two different issues in Mandelbrot set – two alternative definitions which are too technical to describe without a blackboard, but which are both very simple and which I assumed naively to be equivalent. Why did I assume so? Because on the pictures I could not see any difference. Obtaining pictures in one way or another way, I couldn’t tell them apart. Therefore, I assumed they were identical and I went on studying this piece. I found that, again, many interesting observations of which most were very preferred by many other very, very skilled mathematicians. But the idea that these two conditions, definitions, are identical is still open. So there are two definitions in Mandelbrot set, the usual one and another one, and they may theoretically be different. People are getting closer, but have not proven it completely.
Recorded on February 17, 2010
Interviewed by Austin \r\nAllen
How a quest to combine aesthetics with mathematics produced one of math’s most famous, and gorgeous, images: the Mandelbrot set.
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Water may be far more abundant on the lunar surface than previously thought.
- Scientists have long thought that water exists on the lunar surface, but it wasn't until 2018 that ice was first discovered on the moon.
- A study published Monday used NASA's Stratospheric Observatory for Infrared Astronomy to confirm the presence of molecular water..
- A second study suggests that shadowy regions on the lunar surface may also contain more ice than previously thought.
Credits: NASA/Daniel Rutter<p>Still, it's not as if the moon is dripping wet. The observations suggest that a cubic meter of the lunar surface (in the Clavius crater site, at least) contains water in concentrations of 100 to 412 parts per million. That's roughly equivalent to a 12-ounce bottle of water. In comparison, the same plot of land in the Sahara desert contains about 100 times more water.</p><p>But a second study suggests other parts of the lunar surface also contain water — and potentially lots of it. Also publishing their findings in <a href="https://www.nature.com/articles/s41550-020-1198-9#_blank" target="_blank">Nature Astronomy</a> on Monday, the researchers used the Lunar Reconnaissance Orbiter to study "cold traps" near the moon's polar regions. These areas of the lunar surface are permanently covered in shadows. In fact, about 0.15 percent of the lunar surface is permanently shadowed, and it's here that water could remain frozen for millions of years.</p><p>Some of these permanently shadowed regions are huge, extending more than a kilometer wide. But others span just 1 cm. These smaller "micro cold traps" are much more abundant than previously thought, and they're spread out across more regions of the lunar surface, according to the new research.</p>
Credit: dottedyeti via AdobeStock<p>Still, the second study didn't confirm that ice is embedded in micro cold traps. But if there is, it would mean that water would be much more accessible to astronauts, considering they wouldn't have to travel into deep, shadowy craters to extract water.</p><p>Greater accessibility to water would not only make it easier for astronauts to get drinking water, but could also enable them to generate rocket fuel and power.</p><p style="margin-left: 20px;">"Water is a valuable resource, for both scientific purposes and for use by our explorers," said Jacob Bleacher, chief exploration scientist in the advanced exploration systems division for NASA's Human Exploration and Operations Mission Directorate, in a statement. "If we can use the resources at the Moon, then we can carry less water and more equipment to help enable new scientific discoveries."</p>
A study finds 1.8 billion trees and shrubs in the Sahara desert.
- AI analysis of satellite images sees trees and shrubs where human eyes can't.
- At the western edge of the Sahara is more significant vegetation than previously suspected.
- Machine learning trained to recognize trees completed the detailed study in hours.
Why this matters<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDU2MDQ1OC9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYzOTkyODg5NX0.O3S2DRTyAxh-JZqxGKj9KkC6ndZAloEh4hKhpcyeFDQ/img.jpg?width=980" id="3770d" class="rm-shortcode" data-rm-shortcode-id="3c27b79d4c0600fb6ebb82e650cabec0" data-rm-shortcode-name="rebelmouse-image" />
Area in which trees were located
Credit: University of Copenhagen<p>As important as trees are in fighting climate change, scientists need to know what trees there are, and where, and the study's finding represents a significant addition to the global tree inventory.</p><p>The vegetation Brandt and his colleagues have identified is in the Western Sahara, a region of about 1.3 million square kilometers that includes the desert, <a href="https://en.wikipedia.org/wiki/Sahel" target="_blank">the Sahel</a>, and the <a href="https://www.sciencedirect.com/topics/agricultural-and-biological-sciences/subhumid-zones" target="_blank" rel="noopener noreferrer">sub-humid zones</a> of West Africa.</p><p>These trees and shrubs have been left out of previous tabulations of carbon-processing worldwide forests. Says Brandt, "Trees outside of forested areas are usually not included in climate models, and we know very little about their carbon stocks. They are basically a white spot on maps and an unknown component in the global carbon cycle."</p><p>In addition to being valuable climate-change information, the research can help facilitate strategic development of the region in which the vegetation grows due to a greater understanding of local ecosystems.</p>
Trained for trees<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDU2MDQ3MC9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYzNTk5NTI3NH0.fR-n1I2DHBIRPLvXv4g0PVM8ciZwSLWorBUUw2wc-Vk/img.jpg?width=980" id="e02c0" class="rm-shortcode" data-rm-shortcode-id="79955b13661dca8b6e19007935129af1" data-rm-shortcode-name="rebelmouse-image" />
Credit: Martin Brandt/University of Copenhagen<p>There's been an assumption that there's hardly enough vegetation outside of forested areas to be worth counting in areas such as this one. As a result the study represents the first time a significant number of trees — likely in the hundreds of millions when shrubs are subtracted from the overall figure — have been catalogued in the drylands region.</p><p>Members of the university's Department of Computer Science trained a machine-learning module to recognize trees by feeding it thousands of pictures of them. This training left the AI be capable of spotting trees in the tiny details of satellite images supplied by NASA. The task took the AI just hours — it would take a human years to perform an equivalent analysis.</p><p>"This technology has enormous potential when it comes to documenting changes on a global scale and ultimately, in contributing towards global climate goals," says co-author Christian Igel. "It is a motivation for us to develop this type of beneficial artificial intelligence."</p><p>"Indeed," says Brandt says, "I think it marks the beginning of a new scientific era."</p>
Looking ahead and beyond<p>The researchers hope to further refine their AI to provide a more detailed accounting of the trees it identifies in satellite photos.</p><p>The study's senior author, Rasmus Fensholt, says, "we are also interested in using satellites to determine tree species, as tree types are significant in relation to their value to local populations who use wood resources as part of their livelihoods. Trees and their fruit are consumed by both livestock and humans, and when preserved in the fields, trees have a positive effect on crop yields because they improve the balance of water and nutrients."</p><p>Ahead is an expansion of the team's tree hunt to a larger area of Africa, with the long-term goal being the creation of a more comprehensive and accurate global database of trees that grow beyond the boundaries of forests.</p>
Tea and coffee have known health benefits, but now we know they can work together.
Credit: NIKOLAY OSMACHKO from Pexels
- A new study finds drinking large amounts of coffee and tea lowers the risk of death in some adults by nearly two thirds.
- This is the first study to suggest the known benefits of these drinks are additive.
- The findings are great, but only directly apply to certain people.