# The pancake theorem

A previous post mentioned the Borsuk–Ulam theorem. A corollary (for the case of a circle) is the pancake theorem: given two plane regions, there is a cut (line) which divides both in half by area.

The example shows a cut dividing both South America and Africa in half (the theorem doesn’t tell us how to find the cut; I used simulated annealing).

The corollary for the 2-sphere is the ham sandwich theorem: given three 3-dimensional objects, there is a cut (plane) which divides all three in half by volume.

# The Borsuk–Ulam theorem

The mathematical tidbit for today is the Borsuk–Ulam theorem, which states that every continuous function f from the n-dimensional sphere to n-dimensional space must satisfy f(p) = f(−p) for some point p.

In particular, every continuous function f from a 2-dimensional sphere (say, the Earth’s surface) to the plane must satisfy f(p) = f(q) for some antipodal pair of points p and q.

Thus, if we can describe weather by a pair of numbers (say, temperature and rainfall), there must be an antipodal pair of points p and q with the same weather (because two numbers specify a point in the plane).

The maps above (for average maximum temperature) and below (for rainfall) show July weather at various places on Earth, and a pair of points with the same weather is highlighted.

It’s a miracle that it works in this case, of course, because the maps only define temperature and rainfall on the land; I would not have been able to recognise a suitable antipodal pair of points if one or both were at sea.

# The Cannonball Sun, Day 1

In a previous post, I mentioned American university students Will Jones, Kyle Samluk, and Danny Ezzo, who are taking their home-made solar car Pink Skies on the Cannonball Run from New York to Los Angeles. As the map shows, after the first day they are roughly 6% of the way there, in spite of a few hiccups reported on their website and on Instagram. As a bonus, they are also driving through some really beautiful countryside.

Update A: Day 2 (Friday June 25) saw the team driving from Sunbury, PA to Brookville, PA, for a total of 332.6 miles (535 km), roughly 10% of the way across the US in total (map has been edited). Some repairs were needed in Brookville.

Update B: Day 3 (Saturday June 26) saw the team driving into Ohio, roughly 15% of the way across the US in total (map has been edited again). The 24-hour forecast shows rain coming up, however, so it’s time for some strategic planning.

# ASC 2021: one month to go

It is now just over a month until the start of scrutineering in the American Solar Challenge (27 July in Topeka, KS). I have updated my teams list with facts and pictures. As always, the documentation submission progress score gives an idea of which teams are in a good position (most of them, fortunately). A low score indicates that a team, considered as a socio-technical system, is operating in a highly stressed mode. In addition, if a team is building a car without official feedback on design documents, there is an increased chance of failing scrutineering (unless the team has considerable experience with prior ASC events).

There are 15 teams still registered for the race, although I do not expect the Egyptians to attend, and probably not the Taiwanese either. Of the remaining teams, 5 teams (Kentucky 3, Illinois State 17, Missouri S&T 42, Georgia Tech 49, and App State 828) will be racing veteran cars – although with various improvements – and the Brizo of Illini 22 is done. Of the 7 remaining teams building new cars, the Nimbus of MIT 4 is also done:

Nimbus on the track (credit)

In addition, PrISUm 9 have revealed their attractive-looking Cruiser Eliana:

The reveal of Eliana (credit)

Also, Principia 32 is making progress on their Ra XI :

Ra XI aerobody with vinyl wrap (credit)

Western Michigan 30 and Kansas 785 are also still busy with construction, the Freya I of UMNSVP 35 has a rolling chassis and aerobody, and North Carolina 99 is continuing their experiment of gutting a commercial ICE vehicle and turning it into a solar-powered Cruiser. Good luck, one and all!

# Pink Skies Sendoff

In a previous post, I mentioned American university students Kyle Samluk, Will Jones, and Danny Ezzo, who are intending to take a home-made, crowd-funded solar car on the Cannonball Run from New York to Los Angeles, leaving their home in Michigan on 22 June. They, and their car Pink Skies, are being farewelled with ice cream, apparently. Kyle’s favourite solar car team is Agoria, so Belgian beer might have been more appropriate, but I guess that U.S. drinking laws rule that out.

You can follow their progress on their website and on Instagram or Facebook. Good luck, guys!

# The Projective Plane

I have been thinking some more about the famous Möbius strip (see also my post on the Klein bottle). The so-called “Sudanese Möbius Band” in the video above is a Möbius strip stretched so as to make the boundary perfectly circular (it is not named after the country, but after the topologists Sue E. Goodman and Daniel Asimov, and you can purchase a plastic one here).

If we glue two of these Möbius strips together (not actually possible in 3 dimensions), we get a Klein bottle. If we glue one to a disc (also not possible in 3 dimensions), we get a projective plane.

Just for fun, the video below shows a Game of Life glider on the projective plane. The top and bottom of the square are considered to be joined, as are the left and right sides. In both cases, there is a reversal of orientation (a manoeuvre not really possible in 3 dimensions). The glider changes colour as it changes orientation.

Video produced using the R `animation` package.

# The Klein Bottle

I have been thinking about the famous Klein bottle (above). To quote a limerick by Leo Moser:

A mathematician named Klein
Thought the Möbius band was divine.
Said he: “If you glue
The edges of two,
You’ll get a weird bottle like mine.”

Just for fun, here is a Game of Life glider on a Klein bottle. The top and bottom of the square are considered to be joined, so as to form a tube. The ends of the tube (vertical sides of the square) are also joined, but with a reversal of orientation (a manoeuvre not really possible in three dimensions). The glider changes colour as it changes orientation.

Video produced using the R `animation` package.

# ASC 2021: 6 weeks to go

It is now 42 days until the start of scrutineering in the American Solar Challenge. I have updated my teams list. There are 16 teams still registered, although I do not expect the Egyptians or the Puerto Ricans to attend, and perhaps not the Taiwanese either. Of the remaining teams, 5 teams (Kentucky 3, Illinois State 17, Missouri S&T 42, Georgia Tech 49, and App State 828) will be racing veteran cars – although with various improvements. Of the 8 remaining teams building new cars, the Brizo of Illini 22 is done:

Brizo in March (credit)

The Nimbus of MIT 4 has been driving (although I have not seen solar panels), PrISUm 9 will reveal their Cruiser Eliana very soon (on 18 June), and the Freya I of UMNSVP 35 has a rolling chassis:

Freya I in June (credit)

Western Michigan 30, Principia 32, and Kansas 785 are all still busy with construction, while North Carolina 99 is continuing their experiment of gutting a commercial ICE vehicle and turning it into a solar-powered Cruiser. Good luck, one and all!

# The Cannonball Sun

Something a little different in the solar car space today. Will Jones and Kyle Samluk are mechanical engineering students from two different universities in Michigan, with a background in a high school team. Eschewing the mainstream competitions (as many major teams are doing), they have built their own solar car, and between 22 June and 6 July they are planning to drive it the 4,900 km or so of the Cannonball Run from New York to Los Angeles.

The US has been crossed in a solar car before (in 2012), when Bochum’s SolarWorld GT made its round-the-world trip. It clocked up a scenic 6,553 km in just over 50 days, from San Francisco via Dallas to Charleston, SC, requiring frequent charging stops and additional solar panels in the trunk. Will and Kyle, assisted by fellow-student Danny Ezzo, hope to cross the US in less than a third of the time. You can follow their progress on their website and on Instagram or Facebook. They are also raising funds needed for the trip.

Top L: Pink Skies with aerobody but no solar panels, Top R: Pink Skies with solar panels but no aerobody, Bottom L: chassis, showing tadpole wheel configuration, Bottom R: rear (driven) wheel [images from the Pink Skies team].

The chassis of their car Pink Skies is a monocoque made from sheet aluminium, with bulkheads riveted in (it looks more like an Abrams tank than an aircraft, to be honest). The solar panels appear to be originally intended for rooftop use. They make up about a third of the weight, but produce a substantial 2.2 kW of power (roughly double that of a typical ASC or BWSC car).

For energy storage, Will and Kyle have taken the safer LiFePO4 option. The suspension is pretty much missing in action, so the ride is likely to be somewhat bumpy. Top speed appears to be an impressive 110 kph, with a 70 kph cruising speed. Will and Kyle also seem to have done a good job of engineering on-the-fly as testing revealed problems with the initial design. I salute their initiative and their vision and I wish them well as they drive across the American continent.

# And the Trees Clap Their Hands: a book review

And the Trees Clap Their Hands: Faith, Perception, and the New Physics by Virginia Stem Owens (1983, republished 2005, 148 pages)

I recently read And the Trees Clap Their Hands: Faith, Perception, and the New Physics by Virginia Stem Owens. This is a book that addresses the important question “What does it all mean?” with regards to science – what does science really tell us about the world, and how should we respond to that? How can we make sense of it all in a human way?

Virginia Stem Owens was born in 1941 and became a pastor’s wife in the Presbyterian Church (USA), also gaining an MA in English literature from the University of Kansas and a Master of Arts in Religion from the Iliff School of Theology in Denver. She has written numerous books.

And the Trees Clap Their Hands was an enjoyable read, but Owens’ lack of scientific experience is responsible for several flaws in the book. I was a little disappointed at the lack of footnotes and at some glaring errors of fact. For example (p. 92), Owens confuses turbulence (a phenomenon of liquids and gases in the “Old Physics”) with Brownian motion (a microscopic phenomenon resulting from the existence of atoms). I also felt that she skipped over some important things, while not getting others quite right. I should point out, too, that the “New Physics” of the subtitle (relativity and quantum theory) is now roughly a century old. On the other hand, Owens’ writing is lyrically beautiful:

The body I am today came yesterday in a crate of avocados from California. India spins in my tea-drenched blood this morning. Minerals dissolved for millennia in a subterranean aquifer irrigate my interior, passing through the portals of my cell walls, which are themselves filigrees of chemical construction. I am really only a river of dissolute stones, the wash of world-water dammed for a melting moment in the space I call my body, some of it ceaselessly brimming over the spillway and flowing on down drains, into other tributaries, catching in some other body’s pond, until one day the whole structure cracks and buckles, giving up in one great gush its reservoir of mineraled water.” (p. 126)

What does it all mean? Answers have come from Plato, Galileo, and the Bible, to mention just three sources (bust of Plato photographed by Marie-Lan Nguyen)

## Relativity and Time

Since Owens mentions relativity several times, I was surprised to see no mention of spacetime. As Hermann Minkowski wrote in 1923, “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Relativity implies that there is no absolute “present moment” in the Universe, and hence that, of the three theories of time illustrated below, only the Block Universe (Eternalism) can be correct. The philosophical and religious implications of this are huge, and would have been worth discussing.

Three views of time: in Presentism, only the Present exists; in the Growing Block Universe, the past exists as well; and in the Block Universe, also called Eternalism, the universe forms a four-dimensional spacetime “block” in which the future is already written (image © Anthony Dekker)

## Quantum Theory and Observation

One of the key aspects of the century-old “New Physics” is that both matter and light exist as waves and particles. The true reality is a combination of two seemingly contradictory perspectives (this has been used by other writers as an analogy for the relationship between e.g. determinism and free will). The wave function of a particle changes over time according to the Schrödinger equation. It can also undergo wave function collapse, changing from something fuzzy and spread out to something far more definite. How and why the latter phenomenon occurs remains quite mysterious.

One of the classic experiments exploring this involves firing electrons at a detector screen through a double slit:

The double slit experiment (credit: NekoJaNekoJa and Johannes Kalliauer)

Being fuzzy waves, electrons go through both slits at the same time, undergoing interference effects characteristic of waves. They are then detected as particles, with comparatively precise locations:

Individual electrons being detected by a screen after passing through a double slit (credit: Thierry Dugnolle)

Physicists used the word “observation” for the electron being detected by the screen (thus having its wave function collapse). Owens takes it for granted that this means “observation by a human mind” and that the human race therefore, in a sense, creates the universe by observing it. However, this use of the word “observation” is not what most physicists mean (indeed, the universe fairly obviously existed before there were any people). It is, in fact, not clear exactly what constitutes a wave-function-collapsing “observation,” but recent work with quantum computing suggests that it happens even when nobody’s looking (and even when you don’t want it to).

Along the lines of Gary Zukav (whom she cites), Owens tries to build a semi-pantheistic philosophy on top of this – something that is not actually justified by the physics. She also makes a big thing of “the impossibility of isolating the observer from the world” (p. 85), which is not actually a huge problem in the physical sciences, if you know what you’re doing. It’s more of a problem with animal behaviour (as in the famous example below) and an enormous problem in psychology and anthropology.

Konrad Lorenz interacting with geese in the 1930s (Are you my mother?)

## Religion and God

Owens is writing from an explicitly Christian (Presbyterian) perspective, which doesn’t quite sit comfortably with the New Age Zukavian material in this book (there also appears to be some influence from Owen Barfield’s Saving the Appearances: A Study in Idolatry). And while Owens highlights the issue of nonlocality in quantum theory, she does not explore how this might relate to an omnipresent God “behind the scenes.” There are some beautifully written spiritual reflections, but the connection of the religious material to the scientific is somewhat tenuous. Owens seems to want “and all the trees of the field shall clap their hands” to be more than a metaphor, but the physics doesn’t really help with that. In addition, there seems to be some theological confusion regarding the doctrine of the Incarnation.

Goodreads gives this book a score of 3.9. In spite of the beautiful writing and genuine sense of wonder, I can’t go nearly that high (side issue: Goodreads somehow has a cover image with the wrong title!).

And the Trees Clap Their Hands by Virginia Stem Owens: 2½ stars