Visualising Higher Dimensional Spheres

I often try to think of how one might imagine spatial objects that have dimension greater than three. That this problem is nontrivial seems to come from the basic fact that the human mind developed to interpret and analyse the lowly three spatial dimensions that we appear to inhabit–our ancestors never had to navigate 5-dimensional mazes to escape from 4-dimensional tigers and therefore visualising in 3-dimensions was perfectly sufficient for successfully passing on our genes.

Despite this physiological shortcoming, there are various clever ways of trying to get a grip on four-dimensional shapes–see here, for example, for some nice visualisations. I recently learnt of a neat way to construct an n-dimensional sphere (henceforth, n-sphere) from an (n-1)-dimensional sphere by “glueing cones together”. (The relevant article also mentions two other visualisation methods and tries to link the method described below to Dante’s description of the universe in his Divine Comedy.) The basic construction goes like this (try it with n=1 or 2):

  • First fill in the interiors of two (n-1)-spheres to get two n-balls. These n-balls, which will be the two hemispheres of our n-sphere, should be thought of as being made of an infinite number of nested (n-1)-spheres of decreasing radii.
  • Now superpose the two n-balls, identifying their coincident boundaries–this boundary is the equator of our n-sphere.
  • Finally, mentally drag the insides of the two n-balls in opposite directions and away from the equator, such that an interior (n-1)-sphere further from the n-ball’s boundary gets dragged further away.

If you try using this to construct the 4-sphere from two 3-balls, you’ll realise that there’s no fourth dimension in which to drag the insides of the 3-balls. You didn’t expect to be able to actually visualise a 4-sphere all in one go, did you? Nevertheless, it seems like there’s something to be gained from this perspective, even if it won’t prevent you becoming a snack to that 4-dimensional tiger.

Cosmology, Conformism, Conservatism, and Coffee

I’m ripping off a blog post by Peter Coles–which was itself taken from a comment to a post on Sean Carroll’s blog–by pointing out some interesting articles by Avi Loeb that discuss cosmological conformism, cosmological conservatism, and rating the potential success of various research areas. I’ll briefly discuss Loeb’s articles in chronological order.

In this article, Loeb argues that young researchers ought to allocate time to innovative, high-risk, high-reward research areas, as well as to the more conservative mainstream research agendas. Loeb discusses the cultural barriers to this type of research. One of the obvious troubles is that:

Clearly, failure and waste of time are a common outcome of risky projects, just as the majority of venture capital investments lose money (but have the attractive feature of being more profitable than anything else if successful). The fear of losses is sure to keep most researchers away from risky projects, which will attract only those few who are willing to face the strong headwind. Risky projects are accompanied by loneliness. Even after an unrecognized truth is discovered, there is often persistent silence and lack of attention from the rest of the community for a while. This situation contrasts with the nurturing feedback that accompanies a project on a variation of an existing theme already accepted by a large community of colleagues who work on the same topic.

He then gives some examples of low-, medium, and high-risk research, and suggests that astrophysics postdocs should adopt a 50-30-20 distribution of research time to low-, medium-, and high-risk topics, respectively, as opposed to the usual 80-15-5 distribution. Although this article concerns theoretical astrophysics research, I suspect that most of this carries over directly to the rest of theoretical physics.

In the next article, Loeb proposes and discusses the idea of a website run by graduate students that uses publicly available data to assess “the future dividends of various research frontiers”. I quite like this idea in principle. One concern would be that such an assessment would not be any more objective than, say, university rankings, which are frequently criticised for giving seemingly arbitrary weightings to the various factors used in their evaluation metrics. In Loeb’s scheme, this problem is dealt with by using historical data to calculate the weightings that would correctly predict a research areas likelihood of success. Of course, in practice there are enough ill-defined concepts involved that this couldn’t be implemented without bias. As to whether or not this is nevertheless a useful enough idea to implement, I’m undecided.

This article, titled “How to Nurture Scientific Discoveries Despite Their Unpredictable Nature”, suggests that funding agencies should give more support to open research that has no programmatic agenda because the potential benefits from unexpected breakthroughs are so vast that they outweigh the high risk of failure. It’s persuasively written and I completely agree with the main idea: that it’s important to financially support risky innovation, as well as established physics. I’m too ignorant to know whether current practice underfunds research “without programmatic reins tied to specific goals”, but based on what I’ve read in books and on blogs, it’s probably the case. Here’s the final paragraph, mainly because he managed to incorporate a biblical reference:

Progress is not linear in time and sometimes it is even inversely proportional to the contemporaneous level of invested effort. This is because progress rests on lengthy preparatory work which lays the foundation for a potential discovery. Therefore, it is inappropriate to measure success based on the contemporaneous level of allocated resources. Lost resources (time and money) should never be a concern in a culture that is not tied to a specific programmatic agenda, because the long-term benefits from finding something different from what you were seeking could be at an elevated level, far more valuable than these lost resources. This echoes a quote from 1 Samuel (Chapter 9, 20), concerning the biblical story of Saul seeking his lost donkeys. The advice Saul received from Samuel, the person who crowned him as a king after their chance meeting, was simple: “As for the donkeys you lost three days ago, do not worry about them…”.

The most recent article, from May this year, encourages senior scientists to mentor young astrophysics researchers to be bold, creative “architects”, rather than conservative “engineers”. (This reminds me of Lee Smollin’s discussion of “seers” and “craftspeople”.) The opening paragraph sums this up nicely:

Too few theoretical astrophysicists are engaged in tasks that go beyond the refinement of details in a commonly accepted paradigm. It is far more straightforward today to work on these details than to review whether the paradigm itself is valid. While there is much work to be done in the analysis and interpretation of experimental data, the unfortunate by-product of the current state of affairs is that popular, mainstream paradigms within which data is interpreted are rarely challenged. Most cosmologists, for example, lay one brick of phenomenology at a time in support of the standard (inflation+Λ+Cold-Dark-Matter) cosmological model, resembling engineers that follow the blueprint of a global construction project, without pausing to question whether the architecture of the project makes sense when discrepancies between expectations and data are revealed.

The roots of this conformism are obvious:

The unfortunate reality of young astrophysicists having to spend their most productive years in lengthy postdoctoral positions without job security promotes conformism, as postdocs aim to improve their chance of getting a faculty job by supporting the prevailing paradigm of senior colleagues who serve on selection committees.

He goes on to argue why modern cosmology needs architects:

Some argue that architects were only needed in the early days of a field like cosmology when the fundamental building blocks of the standard model, e.g., the inflaton, dark matter and dark energy, were being discovered. As fields mature to a state where quantitative predictions can be refined by detailed numerical simulations, the architectural skills are no longer required for selecting a winning world model based on comparison to precise data. Ironically, the example of cosmology demonstrates just the opposite. On the one hand, we measured various constituents of our Universe to two significant digits and simulated them with accurate numerical codes. But at the same time, we do not have a fundamental understanding of the nature of the dark matter or dark energy nor of the inflaton. In searching for this missing knowledge, we need architects who could suggest to us what these constituents might be in light of existing data and which observational clues should be searched for. Without such clues, we will never be sure that inflation really took place or that dark matter and dark energy are real and not ghosts of our imagination due to a modified form of gravity.

In the original post by Sean Carroll that I mentioned at the start of this post, which is worth reading, Sean plays devil’s advocate to this idea. He ends with this sobering perspective:

Then again, you gotta eat. People need jobs and all that. I can’t possibly blame anyone who loves science and chooses to research ideas that are established and have a high probability of yielding productive results. The real responsibility shouldn’t be on young people to be bomb-throwers; it should be on the older generation, who need to be willing to occasionally take a bomb to the face, and even thank the bomb-thrower for making the effort. Who knows when an explosion might unearth some unexpected treasure?

Phew, it must be time for coffee.

Book Review: Don’t You Have Time to Think?

This book is a collection of the physicist Richard Feynman’s letters, as put together and edited by his daughter Michelle Feynman. There’s also a forward by Timothy Ferris, commentary from Michelle throughout, and six appendices of extra Feynman material.

The letters are roughly ordered temporally, which gives the reader a small amount of insight into most parts of Feynman’s post-teenage life. The first letter is from a 21-year-old Feynman to his mother, Lucille, as he is starting his graduate studies at Princeton University and the last is from an ageing Feynman writing to quell the worries of a father who is anxious about his son’s future. In between, we gain some insight into Feynman’s personal experiences working on the Manhattan project, being a Nobel laureate, and life in general, including a touching personal letter to his deceased first wife, Arline. Some of my favourite excerpts from this book are included below.

If you are a Feynman fan, then you’ll definitely find this book interesting and you’ve probably already read it. You could also gain something from this book if you need reminding that scientists aren’t autonomous beings that exist outside the realm of personal human experience. Indeed, Feynman’s letters offer a glimpse into the real human being that is responsible for the great Feynman legend.

Note: there’s a US version of this book that is published under the less-evocative title: “Perfectly Reasonable Deviations from the Beaten Track”.

From the letter written to the anxious father:

Q: What is it that would make a smart 16-year-old stop for a minute and think… A: Nothing, now, I hope. But to fall in love with a wonderful woman, and to talk to her quietly in the night will do wonders.

In response to the question about what he would do differently in physics if he had his time again:

I would try to forget how I had solved a problem. Then, each time the problem arose, I might solve it in a different way—I wouldn’t be thinking about how I had solved it before.

In the letter to Arline two years after her death:

I find it hard to understand in my mind what it means to love you after you are dead–but I still want to comfort and take care of you–and I want you to love me and care for me. I want to have problems to discuss with you–I want to do little projects with you. I never thought until just now that we can do that together… P.S. Please excuse my not mailing this–but I don’t know your new address.

Book Review: Fearful Symmetry: Is God a Geometer?

“The reason for trying to understand the universe isn’t that we thereby blunder into a new material for coating non-stick frying pans. It’s that we gain insight into our place in the scheme of things, and of just how wonderful and unexpected that scheme can be. The aim of science is not just the manufacture of new toys: it’s the enrichment of the human spirit.”

Fearful Symmetry: Is God a Geometer? by Ian Stewart and Martin Golubitsky is a popular science book from the early 90’s on the subject of symmetry, with an emphasis on the fascinating phenomenon of symmetry breaking. The authors take the reader on an idiosyncratic tour of this subject, discussing cosmology, crystallography, and biology, including, for example, detailed discussions of the Couette-Taylor fluid system, animal gaits, and embryology.  Throughout this book is also contained a sprinkling of sage discussion on the philosophy of science, as the opening quote attests to.

There are many popular books in existence on the topic of symmetry; what makes this book stand out is that the authors have assumed a bit of intelligence on behalf of the reader and this allows Fearful Symmetry to be more sophisticated than your average pop sci book.

However, as a physics student, I was disappointed by what was omitted in the discussion of the standard model (SM) of particle physics. The authors did, to their credit, emphasis the fundamental importance of symmetries in the SM—they even explicitely named the SU(3) gauge symmetry of quantum chromodynamics—and they also mentioned that the electroweak symmetry breaks at low energies, which results in the apparently distinct electromagnetic and weak force. Unfortunately though, there was no mention of the role of symmetry breaking in giving masses to the W and Z bosons and all of the fundamental fermions—a process called the Higg’s mechanism, which is responsible for the weakness of the weak force and the mass of electrons. This, in my biased opinion, is one of the most—if not the most—important examples of symmetry breaking in nature, so it is a shame that it wasn’t included.

Another thing that annoyed me was the statement of the purported mystery that all particles of the same type are identical; this is indeed a deep empirical fact that needs explaining, but it’s not a mystery in the context of the very successful field-based theories of modern particle physics: all particles of the same type are identical (up to a minus sign) because they arise from the same underlying field. There were also some out-dated references to the promise of grand unified theories and string theory, but that’s only because the book was written two decades ago.

Admittedly, I’m being pedantic. It’s only because this book is more intellectual and goes deeper than most non-technical accounts that I hold it to high standards, so this criticism should be taken as a compliment to the rest of the book. Overall, the authors do a good job of explaining the unassuming ubiquity of symmetry and symmetry breaking in the real world, which makes for some fairly interesting reading on an important topic.

Book Review: The Naked Man

This book continues Desmond Morris’ series on Homo sapiens–which includes his well-known “The Naked Ape” and also “The Naked Woman”–with, obviously, this book paying particular attention to the male of the species.

The book is made up of 24 chapters, each devoted to a particular characteristic or body part of the human male and with self-explanatory titles such as “The Hair”, “The Chest”, and “The Penis”. The chapters have a repeating structure–which makes this feel a bit like reading an (unusually engaging) Encyclopaedia at times–in which Morris discusses, amongst other things, the evolutionary context of each body part, gives some cultural perspective, and lists examples of common gestures. Overall this structure makes sometimes for dull reading, but this is compensated for by the many neat ideas and illuminating comments.

For example, in the chapter “The Hands”, Morris states that the handshake might have originated as a way of men assessing one another’s physical strength. This is followed by the insightful remark: “Today some insecure individuals still employ this power grip, crushing the fingers of the people they meet in the forlorn hope that this will impress them.”

In the chapter “The Testicles”, Morris discusses the apparently bizarre phenomenon of external testicles. We learn that their evolutionary origin is probably not primarily as a heat-regulating mechanism, as is commonly believed, but rather that they help to prevent the unfortunate abdominal trauma that could befall any fighting male that housed his testicles internally.

The final chapter, titled “The Preferences”, discusses male sexual preferences, with a particular focus on individuals that are not heterosexual. An interesting comment that Morris makes is that non-breeders, such as monks, priests, and homosexuals, are valuable to an overpopulated humanity. He then says: “This probably explains why, in most advanced countries, where issues of human population are already widely understood, the laws against male homosexuality have recently been relaxed or abandoned…Officially, of course, other reasons are given, such as human rights, privacy laws, sexual liberation, and the rest. But the truth is that, when society makes a major shift in its attitude towards some basic pattern of human behaviour there is usually an underlying factor at work, a factor that has to do with the biological rules of life.” So, could it be that modern society’s evermore-egalitarian attitudes are more a reflection of utilitarian motives, rather than selfless humanity? It’s an interesting idea that I think is probably largely right as a first approximation.

Overall, I’d recommend this book if you’re after a light introduction to male behaviour, so that you can better understand half of those naked apes that you walk amongst; I’m intrigued enough that I now want to read “The Naked Woman”, so that maybe I can find out what’s going on with the other half.

The Unabomber

I had only a passing knowledge of the Unabomber episode when I came across this book in a Phnom Penh bookshop a few weeks ago, but the prospect of finding out more about this mathematician-turned-bomber was enough temptation for me to buy it and find out more. I was given extra impetus to write about this topic when I woke up this morning to the developing news story about the bombing of the Boston marathon, which is vaguely yet disturbingly reminiscent of Ted Kaczynski’s semi-random bombings during the last quarter of last century.

Ted Kaczynski, aka the Unabomber, was a clever mathematician on his way to earning tenure when, two years into an assistant professorship at UC Berkeley, he suddenly quit and became a recluse living in a remote Montana cabin without electricity or running water. After a few years he started mailing bombs to people and organisations. Over a twenty year period he sent a total of 16 bombs across America, killing three people and injuring 23–he seemed to target universities and airlines, hence the name “Unabomber” given to him by the media. In 1995, toward the end of his bombing campaign, his manifesto was published, at his request with the promise to stop sending bombs, by the New York Times and the Washington Post. The gist of his manifesto, Industrial Society and Its Future, and his justification for killing people, is that he thinks that industrialisation inevitably causes individuals to lose their freedom and that a violent revolution is needed to prevent a future in which human life becomes unnatural and meaningless because of our dependence on technology.

The Unabomber’s 35,000 word manifesto is included in the book and I found it interesting enough to read it in full, if only because it gave an insight into how political ideology can motivate an intelligent person to kill people. In fact, there are parts of the manifesto in which it’s surprisingly difficult to disentangle the rational, reasonable ideas from the absurd ones. There’s a particular section titled (something like) “Motivations of Scientists”, which I found intriguing. Kaczynski argues that the common claims by scientists’ that they do science “out of curiosity”, or “to help people”, are fallacious since specialised scientific questions could not satisfy any natural kind of curiosity and that “science marches on blindly, without regard to the real welfare of the human race”. This is meant to degrade science for it is merely a “surrogate activity” that promotes the pursuit of “artificial” goals, rather “real” goals, like hunting and scavenging. My view is that Kaczynski’s argument is mostly bollocks; that curiosity is a legitimate reason to pursue science; and that, in the modern world, science is one of the most meaningful pursuits that one can undertake. Of course, my point of view is exactly what Kaczynski would expect from a budding “Leftist” (to use Kaczynski’s terminology) scientist and therefore it’s probably true that our points of view are irreconcilable.

The book itself is quite unremarkable, probably since it was hastily put together so that it could be published when the story was most topical. I would guess that someone has since written a more eloquent account of the Unabomber episode, but this one does a good enough job of conveying the essential information.

As for the relation of this to the recent bombing of the Boston marathon mentioned earlier, I think it’s plausible that this event is also the result of a rogue individual motivated by a fringe political ideology–although this is completely unfounded speculation.

Zee’s introduction to Feynman’s QED

I recently came across the 2006 edition of Richard Feynman’s wonderful book “QED: The Strange Theory of Light and Matter,” published by Princeton University Press. This book deserves a blog post all to itself, but here I only want to point out the shiny new feature of this most recent edition–surely a publishing scam designed to make me buy the book twice–which is an excellent introduction by Anthony Zee.

Fortunately, it turns out that I don’t need to re-buy the book since Zee has published the introduction online. I recommend reading Zee’s frank and humorous introduction to any aspiring physicists or physics fans out there.