## A New Infinite Series, Convergent and Irrational

Infinite series are a log of func. All kidding aside, you may have heard of the sum of reciprocal squares.

$\sum\limits_{n=1}^{\infty}\frac{1}{{n}^2}=\frac{1}{{1}^2}+\frac{1}{{2}^2}+\frac{1}{{3}^2}+\frac{1}{{4}^2}+\frac{1}{{5}^2}+\frac{1}{{6}^2}+\frac{1}{{7}^2}+\cdots$

The sum of this slowly convergent series is approximately equal to 1.644934. Is there anything special about this number? Italian mathematician Pietro Mengoli (1626-1686) first posed the question in 1644 (published 1650), what is the exact sum of this infinite series? This problem was not solved until 90 years later by Swiss mathematician Leonhard Euler (1707-1783) in 1734 (published 1735). Euler proved that the exact sum is

$\sum\limits_{n=1}^{\infty}\frac{1}{{n}^2}=\frac{{\pi}^2}{6}$

There’s that number pi, the most famous of the irrational numbers, showing up once again in mathematics, ostensibly having nothing at all to do with circles. What’s an irrational number? It is any real number that cannot be expressed as a ratio of two integers. The decimal digits of an irrational number neither terminate nor end in a repeating sequence (e.g. 1/3 = 0.3333… or 9/11 = 0.81818181…).

Determining the exact value of the sum of reciprocal squares infinite series is known as the Basel Problem, named after the hometown of Euler (who solved it) and the Bernoulli family of mathematicians (who were not able to solve it).

My admired colleague in England, Abdul Ahad, has come up with a variant of the sum of reciprocal squares where every third term starting with n = 4 is subtracted rather than added.

$\frac{{\pi}^2}{6} -2\sum\limits_{n=1}^{\infty}\frac{1}{{\left (3n+1 \right )}^2}=\frac{1}{{1}^2}+\frac{1}{{2}^2}+\frac{1}{{3}^2}-\frac{1}{{4}^2}+\frac{1}{{5}^2}+\frac{1}{{6}^2}-\frac{1}{{7}^2}+\cdots$

Ahad has shown that this new series is convergent and sums to an irrational number, approximately equal to 1.40146804. The infinite sum portion of the above expression is approximately equal to 0.12173301 and is also an irrational number. Multiplying by 2 gives us an irrational number, and subtracting from π2/6, which is itself irrational, results in our final result being irrational.

Interestingly, almost all real numbers are irrational, strange as they are.

References

Ahad, Abdul. “A New Infinite Series with Proof of Convergence and Irrational Sum.” Res Rev J Statistics Math Sci, Volume 4, Issue 1 (2018). http://www.rroij.com/open-access/a-new-infinite-series-with-proof-of-convergence-and-irrational-sum.pdf.

## Multiverse

George F. R. Ellis writes in Issues in the Philosophy of Cosmology:

9.2 Issue H: The possible existence of multiverses
If there is a large enough ensemble of numerous universes with varying properties, it may be claimed that it becomes virtually certain that some of them will just happen to get things right, so that life can exist; and this can help explain the fine-tuned nature of many parameters whose values are otherwise unconstrained by physics.  As discussed in the previous section, there are a number of ways in which, theoretically, multiverses could be realized.  They provide a way of applying probability to the universe (because they deny the uniqueness of the universe).  However, there are a number of problems with this concept.  Besides, this proposal is observationally and experimentally untestable; thus its scientific status is debatable.

My 100-year-old uncle—a lifelong teacher and voracious reader who is still intellectually active—recently sent me Max Tegmark’s book Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, published by Vintage Books in 2014. I could not have had a more engaging introduction to the concept of the Multiverse. Tegmark presents four levels of multiverses that might exist. They are

Level I Multiverse: Distant regions of space with the same laws of physics that are currently but not necessarily forever unobservable.

Level II Multiverse: Distant regions of space that may have different laws of physics and are forever unobservable.

Level III Multiverse: Quantum events at any location in space and in time cause reality to split and diverge along parallel storylines.

Level IV Multiverse: Space, time, and the Level I, II, and III multiverses all exist within mathematical structures that describe all physical existence at the most fundamental level.

There seems little question that our universe is very much larger than the part that we can observe. The vast majority of our universe is so far away that light has not yet had time to reach us from those regions. Whether we choose to call the totality of these regions the universe or a Level I multiverse is a matter of semantics.

Is our universe or the Level I multiverse infinite? Most likely not. That infinity is a useful mathematical construct is indisputable. That infinite space or infinite time exists is doubtful. Both Ellis and Tegmark agree on this and present cogent arguments as to why infinity cannot be associated with physical reality. Very, very large, or very, very small, yes, but not infinitely large or infinitely small.

Does a Level II, III, and IV multiverse exist? Tegmark thinks so, but Ellis raises several objections, noted above and elsewhere. The multiverse idea remains quite controversial, but as Tegmark writes,

Even those of my colleagues who dislike the multiverse idea now tend to grudgingly acknowledge that the basic arguments for it are reasonable. The main critique has shifted from “This makes no sense and I hate it” to “I hate it.”

I will not delve into the details of the Level II, III, and IV multiverses here. Read Tegmark’s book as he adroitly takes you through the details of eternal inflation, quantum mechanics and wave functions and the genius and tragic story of Hugh Everett III, the touching tribute to John Archibald Wheeler, and more, leading into a description of each multiverse level in detail.

I’d like to end this article with a quote from Max Tegmark from Mathematical Universe. It’s about when you think you’re the first person ever to discover something, only to find that someone else has made that discovery or had that idea before.

Gradually, I’ve come to totally change my feelings about getting scooped. First of all, the main reason I’m doing science is that I delight in discovering things, and it’s every bit as exciting to rediscover something as it is to be the first to discover it—because at the time of the discovery, you don’t know which is the case. Second, since I believe that there are other more advanced civilizations out there—in parallel universes if not our own—everything we come up with here on our particular planet is a rediscovery, and that fact clearly doesn’t spoil the fun. Third, when you discover something yourself, you probably understand it more deeply and you certainly appreciate it more. From studying history, I’ve also come to realize that a large fraction of all breakthroughs in science were repeatedly rediscovered—when the right questions are floating around and the tools to tackle them are available, many people will naturally find the same answers independently.

References
Ellis, G.F.R., Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Tegmark, Max. Our mathematical universe : my quest for the ultimate nature of reality. New York: Alfred A. Knopf, 2014.

“You passed your exam in many parallel universes—but not in this one.”

## Thank the Sumerians

Over five thousand years ago, the Sumerians in the area now known as southern Iraq appear to have been the first to develop a penchant for the numbers 12, 24, 60 and 360.

It is easy to see why. 12 is the first number that is evenly divisible by six smaller numbers:

12 = 1×12, 2×6, 3×4 .

24 is the first number that is evenly divisible by eight smaller numbers:

24 = 1×24, 2×12, 3×8, 4×6 .

60 is the first number than is evenly divisible by twelve smaller numbers:

60 = 1×60, 2×30, 3×20, 4×15, 5×12, 6×10 .

And 360 is the first number that is evenly divisible by twenty-four smaller numbers:

360 = 1×360, 2×180, 3×120, 4×90, 5×72, 6×60, 8×45, 9×40, 10×36, 12×30, 15×24, 18×20 .

And 360 in a happy coincidence is just 1.4% short of the number of days in a year.

We have 12 hours in the morning, 12 hours in the evening.

We have 24 hours in a day.

We have 60 seconds in a minute, and 60 minutes in an hour.

We have 60 arcseconds in an arcminute, 60 arcminutes in a degree, and 360 degrees in a circle.

The current equatorial coordinates for the star Vega are

α2019.1 = 18h 37m 33s
δ2019.1 = +38° 47′ 58″

Due to precession, the right ascension (α) of Vega is currently increasing by 1s (one second of time) every 37 days, and its declination (δ) is currently decreasing by 1″ (one arcsecond) every 5 days.

With right ascension, the 360° in a circle is divided into 24 hours, therefore 1h is equal to (360°/24h) = 15°. Since there are 60 minutes in an hour and 60 seconds in a minute, and 60 arcminutes in a degree and 60 arcseconds in an arcminute, it follows that 1m = 15′ and 1s = 15″.

Increasingly, you will see right ascension and declination given in decimal, rather than sexagesimal, units. For Vega, currently, this would be

α2019.1 = 18.62583h
δ2019.1 = +38.7994°

Or, both in degrees

α2019.1 = 279.3875°
δ2019.1 = +38.7994°

Even though the latter three forms lend themselves well to computation, I still prefer the old sexagesimal form for “display” purposes, and when entering coordinates for “go to” at the telescope.

There is something aesthetically appealing about three sets of two-digit numbers, and, I think, this form is more easily remembered from one moment to the next.

For the same reason, we still use the sexagesimal form for timekeeping. For example, as I write this the current time is 12:25:14 a.m. which is a more attractive (and memorable) way to write the time than saying it is 12.4206 a.m. (unless you are doing computations).

That’s quite an achievement, developing something that is still in common use 5,000 years later.

Thank the Sumerians!

## Lost in Math: A Book Review

I recently finished reading a thought-provoking book by theoretical physicist Sabine Hossenfelder, Lost in Math: How Beauty Leads Physics Astray. Hossenfelder writes in an engaging and accessible style, and I hope you will enjoy reading this book as much as I did. Do we have a crisis in physics and cosmology? You be the judge. She presents convincing arguments.

The basic premise of Hossenfelder’s book is that when theoretical physicists and cosmologists lack empirical data to validate their theories, they have to rely on other approaches—”beauty”, “symmetry”, “simplicity”, “naturalness“, “elegance”—mathematics. Just because these approaches have been remarkably successful in the past is no guarantee they will lead to further progress.

One structural element that contributes to the book’s appeal is Hossenfelder’s interviews with prominent theoretical physicists and cosmologists: Gian Francesco Giudice, Michael Krämer, Gordon Kane, Keith Olive, Nima Arkani-Hamed, Steven Weinberg, Chad Orzel, Frank Wilczek, Garrett Lisi, Joseph Polchinski, Xiao-Gang Wen, Katie Mack, George Ellis, and Doyne Farmer. And, throughout the book, she quotes many other physicists, past and present, as well. This is a well-researched book by an expert in the field.

I also like her “In Brief” summaries of key points at the end of each chapter. And her occasional self-deprecating, brief, soliloquies, which I find reassuring. This book is never about the care and feeding of the author’s ego, but rather giving voice to largely unspoken fears that theoretical physics is stagnating. And an academic environment hell-bent on preserving the status quo isn’t helping matters, either.

##### Anthropic Principle

Do we live in a universe fine-tuned for life? If so, is it the only possible universe that would support life? Recent work indicates that there may be more than one set of parameters that could lead to a life-supporting universe.

##### Beauty is in the Eye of the Beholder

Is our sense of what is “beautiful” a reliable guide to gaining a deeper understanding of nature? Or does it sometimes lead us astray? We know from history that it does.

In the past, symmetries have been very useful. Past and present, they are considered beautiful

When we don’t have data to guide our theory development, aesthetic criteria are used. Caveat emptor.

##### Experiment and Theory

Traditionally, experiment and observation have driven theory. Now, increasingly, theory drives experiment, and the experiments are getting more difficult, more expensive, and more time consuming to do—if they can be done at all.

##### Inflation

The rapid expansion of the universe at the time of the Big Bang is known as cosmic inflation, or, simply, inflation. Though there is some evidence to support inflation, that evidence is not yet compelling.

##### Mathematics

Mathematics creates a logically consistent universe all its own. Some of it can actually be used to describe our physical universe. What math is the right math?

Math is very useful for describing nature, but is math itself “real”, or is it just a useful tool? This is an ancient question.

##### Memorable Quotations

“I went into physics because I don’t understand human behavior.” (p. 2)

“If a thousand people read a book, they read a thousand different books. But if a thousand people read an equation, they read the same equation.” (p. 9)

“In our search for new ideas, beauty plays many roles. It’s a guide, a reward, a motivation. It is also a systematic bias.” (p. 10)

On artificial intelligence: “Being unintuitive shouldn’t be held against a theory. Like lack of aesthetic appeal, it is a hurdle to progress. Maybe this one isn’t a hurdle we can overcome. Maybe we’re stuck in the foundations of physics because we’ve reached the limits of what humans can comprehend. Maybe it’s time to pass the torch.” (p. 132)

“The current organization of academia encourages scientists to join already dominant research programs and discourages any critique of one’s own research area.” (p. 170)

##### Multiverse

The idea that our universe of just one of a great many universes is presently the most controversial idea in physics.

##### Particles and Interactions

What is truly interesting is not the particles themselves, but the interactions between particles.

##### Philosophy

Physicists and astrophysicists are sloppy philosophers and could stand to benefit from a better understanding of the philosophical assumptions and implications of their work.

##### Physics isn’t Math

Sure, physics contains a lot of math, but that math has traditionally been well-grounded in observational science. Is math driving physics more than experiment and observation today?

##### Quantum Mechanics

Nobody really understands quantum mechanics. Everybody’s amazed but no one is happy. It works splendidly well. The quantum world is weird. Waves and particles don’t really exist, but everything (perhaps even the universe itself) is describable by a probabilistic “wave function” that has properties of both and yet is neither. Then there’s the many-worlds interpretation of quantum mechanics, and quantum entanglement

##### Science and the Scientific Method

In areas of physics where experiments are too difficult, expensive, or impossible to do, some physicists seem to be abandoning the scientific method as the central pillar of scientific inquiry. Faith in beauty, faith in mathematics, faith in naturalness, faith in symmetry. How is this any different than religion?

If scientists can evaluate a theory using other criteria than that theory’s ability to describe observation, how is that science?

##### Stagnation

Some areas of physics haven’t seen any new data for decades. In such an environment, theories can and do run amok.

##### Standard Model (of particle physics)

Ugly, contrived, ad hoc, baroque, overly flexible, unfinished, too many unexplained parameters. These are some of the words used to describe the standard model of particle physics. And, yet, the standard model describes the elementary particles we see in nature and their interactions with extraordinary exactitude.

##### String Theory

String theory dates back at least to the 1970s, and its origins go back to the 1940s. To date, there is still no experimental evidence to support it. String theory is not able to predict basic features of the standard model. That’s a problem.

##### Triple Threat: Crises in Physics, Astrophysics, and Cosmology?

Physics: Sure, the Large Hadron Collider (LHC) at CERN gave us the Higgs boson, but little else. No new physics. No supersymmetry particles. Embarrassments like the diphoton anomaly. Do we need a bigger collider? Perhaps. Do we need new ideas? Likely.

Astrophysics: We’ve spent decades trying to understand what dark matter is, to no avail. No dark matter particles have been found.

Cosmology: We have no testable idea as to what dark energy is. Plenty of theories, though.

See Hossenfelder’s recent comments on the LHC and dark matter in her op-ed, “The Uncertain Future of Particle Physics” in the January 23, 2019 issue of The New York Times.

The book concludes with three appendices:

• Appendix A: The Standard Model Particles
• Appendix B: The Trouble with Naturalness
• Appendix C: What You Can Do To Help

Hossenfelder gives some excellent practical advice in Appendix C. This appendix is divided into three sections of action items:

• As a scientist
• As a higher ed administrator, science policy maker, journal editor, or representative of a funding body
• As a science writer or member of the public

I’m really glad she wrote this book. As an insider, it takes courage to criticize the status quo.

References
Hossenfelder, S., Lost in Math: How Beauty Leads Physics Astray, Basic Books, New York (2018).
Hossenfelder, Sabine. “The Uncertain Future of Particle Physics.” The New York Times 23 Jan 2019. https://www.nytimes.com/2019/01/23/opinion/particle-physics-large-hadron-collider.html.

## Effective Diameter of an Irregularly-Shaped Object

A diameter of a circle in 2D is defined as any straight line segment that intersects the center of the circle with endpoints that lie on the circle.  Since all diameters of a circle have the same length, the diameter is the length of any diameter.

Likewise, a diameter of a sphere in 3D is defined as any straight line segment that intersects the center of the sphere with endpoints that lie on the surface of the sphere, and the diameter is its associated length.

But how do we define the diameter of an irregularly-shaped object such as a typical asteroid or trans-Neptunian object?

For a well-characterized object such as 951 Gaspra—the first asteroid to be photographed up close by a spacecraft—we’ll see the dimensions of the best fitting triaxial ellipsoid given in terms of “principal diameters”.  In the case of Gaspra, that is 18.2 × 10.5 × 8.9 km.

In certain circumstances, however, it would advantageous to characterize an irregularly-shaped object using a single “mean diameter”.  How should we calculate that?

There are two good approaches, provided you have enough information about the object.  The first is to determine the “volume equivalent diameter” which is the diameter of a sphere having the same volume as the asteroid.  This is particularly relevant to mass and density.

For purposes of illustration only, let’s assume Gaspra’s dimensions are exactly the same as its best-fitting triaxial ellipsoid.  If that were true, the volume of Gaspra would be

$V = \frac{{4\pi abc }}{3}$

where V is the volume, and a, b, and c are the principal radii of the triaxial ellipsoid.

Plugging in the numbers 9.1 km, 5.25 km, and 4.45 km (half the principal diameters), we get a volume of 890.5 km3.

The volume equivalent diameter is

$d_{vol} = \left (\frac{6V_{obj}}{\pi } \right )^{1/3}$

where dvol is the volume equivalent diameter, and Vobj is the volume of the object.

Plugging in the volume of 890.5 km3 gives us a volume equivalent diameter of 11.9 km.

The second approach is to determine the “surface equivalent diameter” which is the diameter of a sphere having the same surface area as the asteroid.  This is most relevant to reflectivity or brightness.

Once again using our triaxial ellipsoid as a stand-in for the real 951 Gaspra, we find that the general solution for the surface area of an ellipsoid requires the use of elliptic integrals.  However, there is an approximation that is more straightforward to calculate and accurate to within about 1%:

$S\approx 4\pi\left ( \frac{a^{p}b^{p}+a^{p}c^{p}+b^{p}c^{p}}{3} \right )^{1/p}$

where S is the surface area, p ≈ 1.6075 can be used, and a, b, and c are the principal radii of the triaxial ellipsoid.

Once again plugging in the numbers, we get a surface area of of 478.5 km2.

The surface equivalent diameter is

$d_{sur} = \left (\frac{S_{obj}}{\pi } \right )^{1/2}$

where dsur is the surface equivalent diameter, and Sobj is the surface area of the object.

Plugging in the surface area of 478.5 km3 gives us a surface equivalent diameter of 12.3 km.

You’ll notice that the surface equivalent diameter for 951 Gaspra (triaxial ellipsoid approximation) is 12.3 km which is larger than the volume equivalent diameter of 11.9 km.  The surface equivalent diameter is apparently always larger than the volume equivalent diameter, though I leave it as an exercise for the mathematically-inclined reader to prove that this is so.

References
Herald, David (2018, October 23).  [Online forum comment].  Message
posted to https://groups.yahoo.com/neo/groups/IOTAoccultations/conversations/messages/65158

Thomas, P.C., Veverka, J., Simonelli, D., et al.: 1994, Icarus 107The Shape of Gaspra, 23-26.