# How to use ChatGPT to model exoplanet interiors as a lab for high school students.

Like the introduction of hand calculators into the classroom in the 1970s, ChatGPT offers enormous promise but currently suffers from a variety of negative expectations. Some of the arguments against students using calculators in the 1970s classroom are being used today against ChatGPT. I think there are some applications that work very well if you consider ChatGPT to be an intelligent ‘hand calculator’ in the math and physical science classroom. Here is an example I came up with without much effort!

I. Student Pre-Requisite Knowledge

The following example requires students to work with Scientific Notation, calculating the volume of spheres and shells, and working with the mass:density:volume relationship.

II. Exoplanet Interior Modeling

Astronomers have discovered over 5000 exoplanets orbiting other stars. We call these ‘exoplanets’ so that they don’t get confused with the ‘planets’ in our solar system. From a careful study of these exoplanets, astronomers can figure out how long they take to orbit their stars, their distance from the star, their diameters and their masses. How do they use this information to figure out what the insides of these exoplanets look like? This activity will show how a simple knowledge of mass, volume and density provides the clues!

III. Mass, Density and Volume

Mass, volume and density are related to each other. If two things occupy the same volume but have different masses, the less-massive one will have the lower density.

Density = Mass / Volume.

Example 1: A Prospector had his sample weighed to be 20 grams, and its volume calculated by water displacement and found to be 4 cubic centimeters. If pure gold has a density of 19.3 gm/cc, is his sample actually gold or is it iron pyrite (density 5.0 gm/cc)?

Answer:   Density =  20 gm/4 cc =   5 gm/cc   so it’s iron pyrite or ‘Fools Gold’.

Example 2: A basic principle of physics is that light things of low density float on top of denser things. Why do you have to shake a bottle of salad dressing before you use it?

ChatGPT Query: There are five different liquids mixed together in a bottle. After 10 minutes they sort themselves out. The liquids are:   Olive oil ( 0.92 g/cc ), water ( 1.0  g/cc), molassis ( 1.4 g/cc) , vinegar ( 1.0006 g/cc), honey (1.43 g/cc). From bottom to top, how will the liquids separate themselves?

IV. Designing Mercury with a One-Component Interior Model

Mercury was formed close to the sun where only iron and nickel-rich compounds could condense into a planet. Let’s model Mercury and see what we discover. The actual mass of Mercury is 0.055 times Earth.

Step 1 – Use the formula  for the volume of a sphere V=4/3 pR3 and with a known radius for Mercury of Rm = 2.43×106 meters to get the volume of the planet of V = 6×1019 m3.

Step 2 – Calculate the mass of Mercury for various choices of density. Give the predicted mass for Mercury in multiples of Earth’s mass of 5.97×1024 kg.

Step 3:  Test your knowledge: For a density of 5000 kg/m3 and a radius of 2.43×106 meters, what is the mass of Mercury for these selected values? Give your answer to two significant figures.

Volume = 4/3p (2.43×106meters)3     =  6.0×1019 m3

Mass = 5000 x Volume =  3.0×1023 kg

Mase(Earth units) = 3.0×1023 kg /5.97×1024 kg  = 0.05 times Earth

Use ChatGPT to generate data for plotting. Enter this question into the window:

ChatGPT Query: A sphere has a radius of 2.43×10^6 meters. What is the mass of the sphere if its density is 5000 kg per cubic meter? Express your answer in units of Earth’s mass of 5.97 x 10^24 kg. Give your answer to two significant figures.

Repeat the ChatGPT query four times to generate a mass estimate for densities of 4000, 4500, 5000, 5500 and 6500 kg/m3. Plot these points on a graph of mass versus density and draw a line through the values. Which density gives the best match to the observed mass of Mercury of 0.055 Mearth?  (Answer: about 5500 kg/m3).

V. Designing Mars with Two-Component Models

Now we add two components together for planets that have a high density core and a lower density mantle. These would have formed farther out than the orbit of Mercury but with masses lower than than of Earth. The mathematical model consists of a spherical core with a radius of Rc, surrounded by a spherical shell with an inner radius of Rc and an outer radius of Rp, where Rp is the observed planetary radius.  Mathematically the model looks like this:

M = Dc x 4/3p Rc3 + Dm x 4/3p ( Rp3 – Rc3)

Draw a diagram of the planet’s interior showing Rc and Rp and confirm that this is the correct formula for the total mass of the planet where Dc is the core density, and Dm is the mantle density.

Test Case: An exoplanet is discovered with a mass of 5.97×10^24 kg and a radius of 6,378 kilometers. If the radius of its core is estimated to be Rc = 3,000 km and its core density is 7000 kg/m3, what is the average density of the mantle material?

Vc = 4/3p Rc3  =   4/3p (3000000m)3 = 1.1×1020 m3

Vmantle = 4/3 p Rp3 – Vc  =  1.1×1021 m3 – 1.1×1020 m3 = 9.8×1020 m3.

Solve equation for Dm:

Dm =  ( M – Dc x Vc ) /Vm

Dm = (5.97×1024 – 7000 x 1.1×1020)/9.8×1020 =  5300 kg/m3

Check your answer with ChatGPT using this query. A planet consists of a core region with a radius of Rc and a mantle region extending to the planet’s surface at a radius of Rp. If the planet is a perfect sphere with a radius Rp = 6378 km and Rc = 3000 km, with a total mass of 5.97×10^24 kg, for a core density of 7000 kg/cubic meters, what is the average mantle density? Give the answer to two significant figures.

Now lets use ChatGPT to generate some models and then we can select the best one. We will select a mantle density from three values, 2000, 3000 and 4000 kg/m3. The core density Dc will be fixed at Dc = 9000 kg/m3. We will use the measured radius for Mars of Rp = 3.4×106 meters, and its total mass of Mm = 6.4×1023 kg. We then vary the core radius Rc. We will plot three curves on a graph of Rc versus Mm one for each value of the assumed mantle density. Use this ChatGPT query to generate your data points.

ChatGPT Query: A planet is modeled as a sphere with  a radius of Rp=3.4×10^6 meters. It consists of a spherical core region with a radius of Rc surrounded by a spherical shell with an inner radius of Rc and an outer radius of Rp. The core of the planet has a density of 9000 kg/cubic meters. The radius of the core Rc = 30% of the planet’s radius. If the density of the mantle is 2000 kg/cubic meter, what is the total mass of the planet in multiples of the mass of Earth, which is 5.97×10^24 kg? Give your answer  to two significant figures?

Repeat this query by changing the mantle density and the core radius values and then plot enough points along each density curve to see the trend clearly. An example of an Excel spreadsheet version of this data is shown in this graph:

This graph shows solutions for a two-component mars model where the mantle has three different densities (2000, 3000 and 4000 kg/m3). The average density of mars is 3900 kg/m3. Which core radius and mantle density combinations seem to be a better match for Mar’s total mass of 0.11 Mearth for the given density of the mantle?

VI: Modeling Terrestrial Planets with a three-component interior.

The most general exoplanet model has three zones; a dense core, a mantle and a low-density crust. This is the expected case for Earth-like worlds. Using our Earth as an example, rocky exoplanets have interiors stratified into three layers: Core, mantle, crust.

Core material is typically iron-nickel with a density of   9000 kg/m3

Mantle material is basaltic rock at a density of 4500 kg/m3

Crust is low-density silicate rich material with a density of 3300 kg/m3

The basic idea in modeling a planet interior is that with the three assumed densities, you vary the volume that they occupy inside the exoplanet until you match the actual mass (Mexo) in kilograms and radius (Rexo) in meters of the exoplanet that is observed. The three zones occupy the radii  Rc, Rm, Rp

We will adjust the core and mantle radii until we get a good match to the exoplanet observed total mass and radius. Let’s assume that the measured values for the Super-Earth exoplanet mass is Mp = 2.5xEarth = 1.5×1025 kg,  and its radius is Rp = 1.5xEarth = 9.6×106 meters.

Core Volume  Vcore = 4/3p Rc3

Mantle Volume  Vm = 4/3 p (Rm3 – Rc3)

Crust Volume   Vcrust =  4/3 p (Rp3 – Rm3)

So the total Mass = (9000 Vcore + 4500Vm + 3300Vcrust)/Mp

Rc ,Rm and Rp are the core, mantle and planet radii in meters, and the total mass of the model is given in multiples of the exoplanet’s mass Mp.

Let’s do a test case that we work by hand to make sure we understand what we are doing.

Choose Rc = 30% of Rp and Rm = 80% of Rp. What is the predicted total mass of the exoplanet?

Rc = 0.3 x 9.6×106 meters =  2.9×106 meters.

Rm = 0.8x 9.6×106 meters =  7.7×106 meters.

Then

Vcore =  4/3p (2.9×106)3 = 1.0×1020 m3

Vm =  4/3p ( (7.7×106)3 – (2.9×106)3) =  1.8×1021 m3

Vcrust =  4/3p ((9.6×106)3  – (7.7×106)3) =  1.8×1021 m3

Then  Mass = (9000 Vcore + 4500 Vm + 3300Vcrust)/Mp

Mass = (9×1023 kg + 8.1×1024 kg + 5.9×1024 kg)/1.5×1025 kg  =  1.0 Mp

Now lets use ChatGPT to generate some models from which we can make a choice.

Enter the following query into ChatGPT to check your answer to the above test problem.

ChatGPT Query: A spherical planet with a radius of Rp consists of three interior zones; a core with a radius of Rcore, a mantle with an inner radius of Rc and an outer radius of Rm,  and a crust with an inner radius of Rm and an outer radius of Rp=9.6×10^6 meters. If the density of the core is 9000 kg/m^3, the mantle is 4500 kg/m^3 and the crust is 3300 kg/m^3, What is the total mass of the planet if Rc = 30% of Rp and Rm = 80% of Rp? Give your answer for the planet’s total mass in multiples of the planet’s known mass of 1.5×10^25 kg, and to two significant figures.

Re-run this ChatGPT query but change the values for the mantle radius Rm and core radius Rc each time. Plot your models on a graph of   Rc versus the calculated mass Mp on curves for which Rm is constant. An example of this plot is shown in the excel spreadsheet plot below.

For example, along the black curve we are using Rm=0.8. At Rc = 0.5 we have a model where the core extends to 50% of the radius of the exoplanet .The mantle extends to 80% of the radius, and so the crust occupies the last 20% of the radius to the surface. With densities of 9000, 4500 and 3300 kg/m3 respectively, the Y-axis predicts a total mass of about 1.1 times the observed mass of the exoplanet (1.00 in these units). With a bit of fine-tuning we can get to the desired 1.00 of the mass.    But what about the solution at (0.3, 1.00) ? In fact, all of the solutions along the horizontal line along y = 1.00 are mathematically valid.

Question 1: The exoplanet is located close to its star where iron and nickel can remain in solid phase but the lower density silicates remain in a gaseous phase. Which of the models favors this location at formation?

Answer: The exoplanet should have a large iron/nickel core and not much of a mantle or crust. This favors solutions on the y=1.00 line to the right of x=0.5.

Question 2: The exoplanet is located far from its star where it is cool enough that silicates can condense out of their gas phase as the exoplanet forms. Which of the models favor this location?

Answer: The exoplanet will have a small iron/nickel core and a large mantle and crust. This favors models to the left of x= 0.5.

So here you have some examples for how ChatGPT can be used as an intelligent calculator once the students understands how to use the equations and is able to explain why they are being used for a given modeling scenario.

I would be delighted to get your responses and suggestions to this approach . Just include your comment in the Linkedin page where I have posted this idea.

# Thinking Visually

Look at the two images  for a few minutes and let your mind wander.

What impressions do you get from the patterns of light and dark? If I were to tell you that the one at the top is a dark nebula in the constellation Orion, and the one on the bottom is a nebula in the Pleiades star cluster, would that completely define for you what you are experiencing…or is there something more going on?

Chances are that, in the top image you are seeing what looks like the silhouette of the head and shoulders of some human-like figure being lit from behind by a light. You can’t quite put your finger on it, but the image seems vaguely mysterious and perhaps even a bit frightening the more you stare at it.

The image on the bottom evokes something completely different. Perhaps you are connecting the translucence and delicacy with some image of a shroud or silken cloak floating in a breeze. The image seems almost ghost-like in some respects…spiritual

But of course this is rather silly” you might say. “These are interstellar clouds, light-years across and all we are doing is letting our imaginations wander which is not a very scientific thing to do if you want to understand the universe.” This rational response then tempts you to reach for your mouse and click to some other page on the web.

What has happened in that split second is that a battle has been fought between one part of your brain and another. The right side of your brain enjoys looking at things and musing over the patterns that it finds there. Alas, it cannot speak because the language centers of the brain live in the left cerebral hemisphere, and it is here that rules of logic and other ‘scientific’ reasoning tools exist. The left side of your brain is vocal, and talking to you right now. It gets rather upset when it is presented with vague patterns because it can’t understand them and stamp them with a definite emotion the way the right hemisphere can. So it argues you into walking away from this challenge of understanding patterns.

If you can suspend this indignation for a moment or two, you will actually find yourself thinking about space in a way that more nearly resembles how a scientist does, though even some scientists don’t spend much time thinking about space. This indifference has begun to change during the last 20 years, and we are now in the midst of a quiet revolution.

There are three child-like qualities that make for a successful scientist:

Curiosity. This is something that many people seem to outgrow as they get older, or if they maintain it as adults, it is not at the same undiluted strength that it was when they were a child.

Imagination. This is something that also wanes with age but becomes an asset to those that can hang on to even a small vestige of it. It is what ‘Thinking out of the box’ is all about.

Novelty. As a child, everything is new. As an adult we become hopelessly jaded about irrelevant experiences like yet another sunset, yet another meteor shower, yet another eclipse. In some ways we develop an aversion for new experiences preferring the familiarity of the things we have already experienced.

Remember, the right brain uses ALL sensory inputs to search for patterns and to understand them. It even uses imaginary information, dreams, and other free-forms to decode what it is experiencing.

My book ‘Exploring Quantum Space’ is a guidebook that will give you some of the mental tools you will need to make sense of one of the greatest, and most subtle, discoveries in human history. Space, itself, is far from being ‘nothing’ or merely a container for matter to rattle around within. It is a landscape of hidden patterns and activity that shapes our universe and our destiny. You cannot understand it, or sense the awe and mystery of its existence, by simply reading words and following a logical exposition of ‘ifs and thens’. You also have to experience it through evocative imagery and imagination. Space is such a different medium from anything we have ever had to confront, intellectually, that we need to employ a different strategy if we wish to understand it in a personal way. Once we do this, we will be reconnected with that sense of awe we feel each time we look at the night sky.

My next blog about Nothing introduces some of the other ideas and techniques that scientists use to think about the impossible!

Looking back at the millennia of model building and deduction that has occurred, not a century has gone by when the prevailing opinion hasn’t been that a perfectly empty vacuum is impossible.

Aristotle’s Aether blends seamlessly into the 19th century Ether. In this century, overlapping quantum waves and virtual particles have finally taken root as the New Ether, though it is now infinitely more ephemeral than anything Aristotle or Maxwell could have imagined. We have also seen how the Atomist School of ancient Greece reached its final vindication in the hands of 19th century scientists such as Boltzman. By the 20th century, the Atomist’s paradigm has even been extended to include not just the graininess of matter, but the possible quantum graininess of the vacuum and space itself. In the virtual particles that animate matter, we finally glimpse the world which Heinrich Hertz warned us about nearly a century ago when he said that we would eventually have to reach some accommodation with “invisible confederates” existing alongside what we can see, to make our whole model of reality more logically self-consistent.

Even by the start of the 21st Century, we have reached this accommodation only by shrugging our shoulders and honestly admitting that there are things going on in the world that seem to defy human intuition. What impresses me most about the evolution of our vision of the vacuum is that the imagery we find so potent today is actually in some sense thousands of years old.

It is difficult to imagine that humans would be drawn to the same understanding of physics and astronomy that we now enjoy if our brains had been wired only slightly differently. Without sight and mobility we could not form the slightest notion of 3-D space and geometry. This is what Kant spoke about, what Henri Poincare described at great length without the benefit of 20th century neuroscience, and what Jacob Bronowski described in his book The Origins of Knowledge and Imagination with the benefit of such knowledge. But the object of science is more than just making sense of our senses. It must also guide us towards a deeper understanding of the physical world. This understanding must be self-consistent, and independent of whether we are sensorially or neurologically handicapped. Mathematics as the premier language of physical model building, seems uniquely suited to providing us with an understanding of the physical world. Mathematics lets us see the world in a way that all of the other human languages do not.

If our mathematical understanding of nature is a product of mental activity, and this activity can be physically affected by the hard-wiring of our brain, how do we arrive at a coherent model of the physical world? Can we see in this process any explanation for why certain ideas in physics appear to be so historically tenacious?

It is commonly believed that in order for mathematics and the underlying logic to exist, at the very least a conscious language must be pre-existent to support it. This is the point of view expressed by Benjamin Whorf. But the thoughtful reflections by individuals such as Einstein, Feynman and Penrose point in a very different direction. Einstein once wrote a note to Jaques Hadamard prompted by Hadamard’s investigation of creative thinking,

“…The words of language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements of thought are certain signs ( symbols ) and more or less clear images which can be voluntarily reproduced and combined…The above mentioned elements are, in my case, of visual and some muscular type…”

Roger Penrose echoes some of this same description in his book, The Emperor’s New Mind,

“…Almost all my mathematical thinking is done visually and in terms of non-verbal concepts, although the thoughts are quite often accompanied by inane and almost useless verbal commentary such as ‘that thing goes with that thing and that thing goes with that thing’..”

Freeman Dyson, one of the architects of modern QED had this to say about how Feynman did his calculations,

“…Dick was using his own private quantum mechanics that nobody else could understand. They were getting the same answers whenever they calculated the same problem…The reason Dick’s physics was so hard for ordinary people to grasp was that he did not use equations…Dick just wrote down the solutions out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the pictures gave him the solutions directly with a minimum of calculation…It was no wonder that people who had spent their lives solving equations were baffled by him. Their minds were analytical; his was pictorial…”

In many instances, the conversion of abstract thinking into conventional language is seen as a laborious, almost painful process. Often words are inadequate to encompass the subtleties of the non-verbal, abstract ideas and their interrelationships. According to Penrose,

“I had noticed, on occasion, that if I have been concentrating hard for a while on mathematics and someone would engage me suddenly in conversation, then I would find myself almost unable to speak for several seconds”

In fact, abstract thinking is often argued to be a right-hemisphere function. Visual or pattern-related thinking and artistic talents are frequently coupled to this hemisphere, and since the language centers are in the left-hemisphere, with such a disconnect between language and abstract thinking, there is little wonder that theoreticians and artists find themselves tongue-tied in explaining their ideas, or are inclined to report that their work is non-verbal.

So the creation of sophisticated physical theories may involve a primarily non-verbal and visual-symbolic thinking processes, often manipulating patterns and only later, with some effort of will, translating this into spoken language or fleshing out the required mathematical details. Could this be why scientists, and artists for that matter have such difficulty in explaining what they are thinking to the rest of the population? Could this be why ancient philosophers managed to land upon archetypes for their Creation legends that seem familiar to us in the 20th century? The symbols that are used appear disembodied, and no amount of word play can capture all of the nuances and motivations that went into a particular interpretive archetypes, and make them seem compelling to the non-mathematician or non-artist. Feynman once wrote about the frustrating process of explaining to the public what goes on in nature,

“…Different people get different reputations for their skill at explaining to the layman in layman’s language these difficult and abstruse subjects. The layman then searches for book after book in the hope that he will avoid the complexities which ultimately set in, even with the best expositor of this type. He finds as he reads a generally increasing confusion, one complicated statement after another,… all apparently disconnected from one another. It becomes obscure, and he hopes that maybe in some other book there is some explanation…but I do not think it is possible, because mathematics is NOT just another language. Mathematics is a language plus reasoning…if you do not appreciate the mathematics, you cannot see, among the great variety of facts, that logic permits you to go from one to the other…”

If this is the mental frame used by some physicists to comprehend physics, it is little wonder that a great chasm exists between the lay person and the physicist in explaining what is going on. The task that even a physicist such as Freeman Dyson had in translating Feynman’s diagrammatic techniques into mathematical symbology, seems even more challenging knowing that Feynman may have had a whole other perspective on visualization via his apparent color-symbol synthesia. The equations below are the current best mathematical expression for the Standard Model in physics, which describes all known particles and fields excepting gravity.

Another feature of thinking that separates scientists and artists from everyone else seems to be the plasticity of the thinking process itself. Scientists flit from one idea to another until they arrive at a model that best explains the available data, although scientists can also get rooted to particular perspectives that are difficult to forget after decades of inculcation. The general adult population prefers a more stable collection of ideas and ‘laws’ which it can refer to over a lifetime.

Where does this all leave us?

The vacuum has been promoted to perhaps the most important clue to our own existence. The difficulty is that we lack a proper Rosetta Stone to translate the various symbolisms we use to describe it. The clues that we do have are scattered among a variety of enigmatic subjects which strain at our best intellectual resources to understand how they are linked together. Could it be that we are lacking an even more potent symbolic metaphor, and an internal non-verbal language, to give it life? Where would such a thing come from?

If we take our clue from how ideas in physics have emerged in the past, the elements of the new way of thinking may be hidden in some unexpected corner of nature. We may find an analogy or a metaphor in our mundane world which, when mixed with mathematical insight, may take us even closer to understanding gravity, spacetime and vacuum. It is no accident that string theory owes much of its success because it asks us to think about quantum fields as ordinary strings operating in an exotic mathematical setting. It is exciting to think that the essential form of the Theory of Everything could be this close to us, perhaps even lurking in a pattern we see, and overlook, in our everyday lives.

Much of this symbolic process may be performed sub-consciously, and only the form of dreams, insights or hunches seem to bring them into consciousness when the circumstances are appropriate. It is, evidently, the non-verbal and unconscious right hemisphere which experiences these ideas. Is there a limit to this process of symbolic thinking? At least a dozen times this century, physicists have had to throw up their hands over what to make of certain features of the world: the collapse of the wave function; quantum indeterminacy; particle/wave dualism; cosmogenesis. Some of these may eventually find their explanation at the next level of model building. Others such as the meaning of quantum indeterminacy and particle/wave dualism, seem to be here to stay.

In working with these contradictions, the human mind prefers the avenue of denial, you can almost hear your inner voice saying “Aw come on, quantum mechanics just can’t be that weird!” or a state of anxiety as the two hemispheres try to fabricate conflicting world models. Little wonder that we have particle/wave duality, the seeming schism between matter and energy, and a whole host of other ‘polar’ ideas in physics, as two separate minds try to resolve the universe into one model or another with the left one preferring time ordered patterns, and the right one, spatial patterns.

It is hard to believe that our brains can control what we experience of the objective world, but we need only realize that the brain actually blindsides us in a variety of subtle ways, from seeing a wider sensory world. The object of science, however, is to discern the shapes of objective laws in a way that gets to the universal elements of nature that are not coupled to a particular kind of brain circuitry. It doesn’t matter if all scientists have anasognosia and see the world differently in some consistent way, what counts is that they must still live by the laws of motion dictated by gravity and quantum mechanics.

Nils Bohr believed atoms are not real in the same sense as trees. The quantum world really does represent a different kind of reality than our apparently naive understanding of macroscopic reality implies. This being the case, we must first ask to what extent fields and the denizens of the quantum vacuum can be represented by any analogy drawn from the macroworld? We already know that the single most important distinguishing characteristic of atomic particles is their spin; far more so than mass or charge. Yet unlike mass and charge, quantum mechanical spin has ABSOLUTELY no analog in the macroscopic world. Moreover, fundamental particles cannot be thought of as tiny spheres of charged matter located at specific points in space. They have no surface, and participate in an infernal wave-like dance of probability, at least when they are not being observed. Yet despite this warning, we feel comfortable that we understand something about what reality is at this scale, in the face of these irreconcilable differences between one set of mental images and what experiments tell us over and over again. What is the true nature of the vacuum? How did the universe begin? I suspect we will not know the answer to these questions in your lifetime or mine, perhaps for the same reason that it took 3000 years for geometers to ‘discover’ non-Euclidean geometry.

At the present time we are faced with what may amount to only a single proof of the parallel-line postulate, unable to see our way through to another way of looking at the proof. There is also the very real worry that some areas of nature may require modalities of symbolic thinking beyond the archetypes that our brains are capable of providing as a consequence of their neural hard-wiring. Today, we have quantum field theory and its tantalizing paradoxes, much as the ancient geometers had their parallel-line postulate. We, like they, scratch the same figures in the sand over and over again, hoping to see the glimmerings of a new world view appearing in the shifting sands. At a precision of one part in a trillion, our quantum theories work too well, and seem to provide few clues to the new direction we must turn to see beyond them.

The primary arbiters we have at our disposal to decide between various interpretive schemes, experimental data, are not themselves in unending supply as the abrupt cancellation of the U.S. Superconducting Super Collider program in 1989 showed. It was replaced by the CERN Large Hadron Collider shown above, but even the LHC may not be large enough to access the new physics we need to explore to further our theories and understanding.

Whatever answers we need seem to be hidden, not in the low- energy world accessible to our technology, but at vastly higher energies well beyond any technology we are likely to afford in the next few centuries. It is easy to provide a jet plane with an energy of 100 billion billion billion volts — its energy of motion at a speed of a few hundred miles per hour, but it is beyond understanding how to supply a single proton or electron with the same energy. On the other hand, our internal symbolic thinking seems to lead us to similar interpretative schemes, and unconscious dualities which may only be a reflection of our own neural architecture, which we all share, and which has remained essentially unchanged for millennia. We visualize the vacuum in the same way as the Ancients did because we are still starting from the same limited collection of internal imagery. At least for some general problems, we seem to have hit a glass ceiling for which our current style of theory building seems to lead us to a bipolar and contradictory world populated by various dualities: matter/energy, space/time, wave/particle. When we finally do break through to a new kind of reality in our experiments, would we be able to recognize this event? Will our brains filter out this new world and show us only the ghostly shadows of contradictory archetypes cast upon the cave wall?

We have seen that many schemes have been offered for describing the essential difference between matter and empty space; many have failed. Theoreticians since Einstein have speculated about the geometric features of spacetime, and the structure of electrons and matter for decades. The growing opinion now seems to be that, ultimately, only the properties of space such as its geometry or dimensionality can play a fundamental role in defining what matter really is. In a word, matter may be just another form of space. If the essence of matter is to be found in the geometric properties of ’empty’ space, our current understanding of space will not be sufficient to describe all of matter’s possible aspects.

# Oops…One more thing!

After writing thirteen essays about space, I completely forgot to wrap up the whole discussion with some thoughts about the Big Picture! If you follow the links in this essay you will come to the essay where I explained the idea in more detail!

Why did I start these essays with so much talk about brain research? Well, it is the brain, after all, that tries to create ideas about what you are seeing based on what the senses are telling it. The crazy thing is that what the brain does with sensory information is pretty bizarre when you follow the stimuli all the way to consciousness. In fact, when you look at all the synaptic connections in the brain, only a small number have anything to do with sensory inputs. It’s as though you could literally pluck the brain out of the body and it would hardly realize it needed sensory information to keep it happy. It spends most of its time ‘taking’ to itself.

The whole idea of space really seems to be a means of representing the world to the brain to help it sort out the rules it needs to survive and reproduce. The most important rule is that of cause-and-effect or ‘If A happens then B will follow’. This also forms the hardcore basis of logic and mathematical reasoning!
But scientifically, we know that space and time are not just some illusion because objectively they seem to be the very hard currency through which the universe represents sensory stimuli to us. How we place ourselves in space and time is an interesting issue in itself. We can use our logic and observations to work out the many rules that the universe runs by that involve the free parameters of time and space. But when we take a deep dive into how our brains work and interfaces with the world outside our synapses, we come across something amazing.

The brain needs to keep track of what is inside the body, called the Self, and what is outside the body. If it can’t do this infallibly, it cannot keep track of what factors are controlling its survival, and what factors are solely related to its internal world of thoughts, feelings, and imaginary scenarios. This cannot be just a feature of human brains, but has to also be something that many other creatures also have at some rudimentary level so that they too can function in the external world with its many hazards. In our case, this brain feature is present as an actual physical area in the cerebral cortex. When it is active and stimulated, we have a clear and distinct perception of our body and its relation to space. We can use this to control our muscles, orient ourselves properly in space, walk and perform many other skills that require a keen perception of this outside world. Amazingly, when you remove the activity in this area through drugs or meditation, you can no longer locate yourself in space and this leads to the feeling that your body is ‘one’ with the world, your Self has vanished, and in other cases you experience the complete dislocation of the Self from the body, which you experience as Out of Body travel.

What does this have to do with space in the real world? Well, over millions of years of evolution, we have made up many rules about space and how to operate within it, but then Einstein gave us relativity, and this showed that space and time are much more plastic than any of the rules we internalized over the millennia. But it is the rules and concepts of relativity that make up our external world, not the approximate ‘common sense’ ideas we all carry around with us. Our internal rules about space and time were never designed to give us an accurate internal portrayal of moving near the speed of light, or functioning in regions of the outside world close to large masses that distort space.

But now that we have a scientific way of coming up with even more rules about space and time, we discover that our own logical reasoning wants to paint an even larger picture of what is going on and is happy to do so without bothering too much with actual (sensory) data. We have developed for other reasons a sense of artistry, beauty and aesthetics that, when applied to mathematics and physics, has taken us into the realm of unifying our rules about the outside world so that there are fewer and fewer of them. This passion for simplification and unification has led to many discoveries about the outside world that, miraculously, can be verified to be actual objective facts of this world.

Along this road to simplifying physics, even the foundations of space and time become players in the scenery rather than aloof partners on a stage. This is what we are struggling with today in physics. If you make space and time players in the play, the stage itself vanishes and has to somehow be re-created through the actions of the actors themselves .THAT is what quantum gravity hopes to do, whether you call the mathematics Loop Quantum Gravity or String Theory. This also leads to one of the most challenging concepts in all of physics…and philosophy.

What are we to make of the ingredients that come together to create our sense of space and time in the first place? Are these ingredients, themselves, beyond space and time, just as the parts of a chain mail vest are vastly different than the vest that they create through their linkages? And what is the arena in which these parts connect together to create space and time?

These questions are the ones I have spent my entire adult life trying to comprehend and share with non-scientists, and they lead straight into the arms of the concept of emergent structures: The idea that elements of nature come together in ways that create new objects that have no resemblance to the ingredients, such as evolution emerging from chemistry, or mind emerging from elementary synaptic discharges. Apparently, time and space may emerge from ingredients still more primitive, that may have nothing to do with either time or space!

You have to admit, these ideas certainly make for interesting stories at the campfire!

Check back here on Monday, December 26 for the start of a new series of blogs on diverse topics!

# Is Space Real?

I take a walk to the store and can’t help but feel I am moving through something that is more than the atmosphere that rushes by my face as I go. The air itself is contained within the boundaries of the space through which I pass. If I were an astronaut in the vacuum of outer space, I would still have the sense that my motion was through a pre-existing, empty framework of 3-dimensions. Even if I were blind and confined to a wheelchair, I could still have the impression through muscular exertion that I was moving through space to get from my kitchen to my living room ‘over there’. But what is space as a physical thing? Of all the phenomena, forces and particles we study, each is something concrete though generally invisible: a field; a wave; a particle. But space, itself, seems to be none of these. WTF!

Way back in the early 1700s, Sir Isaac Newton proposed that space was an ineffable, eternal framework through which matter passed. It had an absolute and immutable nature. Its geometry pre-existed the matter that occupied it and was not the least bit affected by matter. A clever set of experiments in the 20th century finally demonstrated rather conclusively that there is no pre-existing Newtonian space or geometry ‘beneath’ our physical world. There is no absolute framework of coordinates within which our world is embedded. What had happened was that Albert Einstein developed a new way of thinking about space that essentially denied its existence!

Albert Einstein’s relativity revolution completely overturned our technical understanding of space and showed that the entire concept of dimensional space was something of a myth. In his famous quote he stressed that We entirely shun the vague word ‘space’ of which we must honestly acknowledge we cannot form the slightest conception. In the relativistic world we live in, space has no independent existence. “…[prior-geometry] is built on the a priori, Euclidean [space], the belief in which amounts to something like a superstition“. So what could possibly be a better way of thinking about space than the enormously compelling idea that each of us carries around in our brains, that space is some kind of stage upon which we move?

To understand what Einstein was getting at, you have to completely do away with the idea that space ‘is there’ and we move upon it or through it. Instead, relativity is all about the geometry created by the histories (worldlines) of particles as they move through time. The only real ‘thing’ is the collection of events along each particle’s history. If enough particles are involved, the histories are so numerous they seem like a continuous space. But it is the properties of the events along each history that determine the over-all geometry of the whole shebang and the property we call ‘dimension’, not the other way around.

This figure is an example where the wires (analogous to worldlines) are defining the shape and contours of a dimensional shape. There is nothing about the background (black) space that determines how they bend and curve. In fact, with a bit of mathematics you could specify everything you need to know about the surface of this shape and from the mathematics tell what the shape is, and how many dimensions are required to specify it!

Princeton University physicist Robert Dicke expressed it this way, “The collision between two particles can be used as a definition of a point in [space]…If particles were present in large numbers…collisions could be so numerous as to define an almost continuous trajectory…The empty background of space, of which ones knowledge is only subjective, imposes no dynamical conditions on matter.”

What this means is that so long as a point in space is not occupied by some physical event such as the interaction point of a photon and an electron, it has no effect on a physical process ( a worldline) and is not even observable. It is a mathematical ‘ghost’ that has no effect on matter at all. The interstitial space between the events is simply not there so far as the physical world based upon worldlines is concerned. It is not detectable even by the most sophisticated technology, or any inventions to come. It does not even supply something as basic as the ‘dimension’ for the physical world!

We should also be mindful of another comment by Einstein that “…time and space are modes by which we think and not conditions in which we live“. They are free creations of the human mind, to use one of Einstein’s own expressions. By the way, the 18th century philosopher Immanuel Kant also called the idea of ‘space’ an example of a priori knowledge that we are born with to sort out the world, but it is not necessarily a real aspect of the world outside our senses.

Like a spider web, individual and numerous events along a worldline define the worldline’s shape, yet like the spider web, this web can be thought of as embedded in a larger domain of mathematically-possible events that could represent physical events…but don’t. The distinction between these two kinds of points is what Einstein’s revolutionary idea of relativity provided physicists, and is the mainstay of all successful physical theories since the 1920s. Without it, your GPS-enabled cell phones would not work!

So what are these events? Simply put, according to Physicist Lee Smolin, they are exchanges of information, which are also the interaction points between one particle’s worldline and another particle’s world line. If you think at the atomic level, each time a particle of light interacts with (collides or is emitted by) an electron it generates an event. These events are so numerous the electron’s worldline looks like a continuous line with no gaps between the events. So the shape of one worldline, what we call its history, is a product of innumerable interactions over time with the worldlines of all other objects (photons etc) to which it can be in cause-and-effect contact.

Even though this new idea of space being a myth has gained enormous validity among physicists over the last century, and I can easily speak the language of relativity to describe it, personally, my mind has a hard time really understanding it all. I also use the mathematical theory of quantum mechanics to make phenomenally accurate predictions, but no Physicist really understands why it works, or what it really means.

Next time I want to examine how the history of a particle is more important than the concept of space in Einstein’s relativity, and how this explains the seeming rigidity of the world you perceive and operate within.

Check back here on Thursday, December 15 for the next installment!

# Seeing with Mathematics

Our brain uses sensory data to sift for patterns in space and time that help us create a mental model of the world through which we can navigate and stay alive. At some point, this model of the external world becomes our basis for thinking symbolically and mathematically about it.

Mathematics is an amazingly detailed, concise and accurate way of examining the world to state the logical relationships we find there, but many physicists and mathematicians have been astonished about why this is the case. The physicist Eugene Wigner wrote an article about this in 1960 titled ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. In fact, since the enormous successes of Sir Isaac Newton in mathematically explaining a host of physical phenomena, physicists now accept that mathematics actually serves as a microscope (or telescope!) for describing things and hidden relationships we cannot directly experience. This amazing ability for describing relationships in the world (both real and imagined!) presents us with a new problem.

Mathematics is a symbolic way of describing patterns our world, and sometimes these symbolically-defined descriptions actually look like the things we are studying. For example, the path of a football is a parabola, but the equation representing its path, y(x), is also that of a parabolic curve drawn on a piece of paper. But what happens when the mathematical description takes you to places where you cannot see or confirm the shape of the object?

Mathematics is a tool for understanding the world and symbolically stating its many logical interconnections, but the tool can sometimes be mistaken for the thing itself. Here is a very important example that comes up again and again when physicists try to ‘popularize’ science.

In the late-1940s, physicist Richard Feynman created a new kind of mathematics for making very precise calculations about how light (photons) and charged particles (such as electrons) behave. His famous ‘Feynman Diagrams’ like the one below, are very suggestive of particles moving in space, colliding, and emitting light. This diagram, with time flowing from left to right, shows a quark colliding with an anti-quark, which generates a photon that eventually produces an electron and anti-electron pair.

The problem is that this is not at all a ‘photograph’ of what is actually happening. Instead, this is a tool used for setting up the problem and cranking through the calculation. Nothing more. It is a purely symbolic representation of the actual world! You are not supposed to look at it and say that for the solid lines, ‘particles are like billiard balls moving on a table top’ or that the photon of light they exchange is a ‘wiggly wave traveling through space’. What these objects are in themselves is completely hidden behind this diagram. This is a perfect example of what philosopher Immanuel Kant was talking about back in the 1700s. He said that there is a behind-the-scenes world of noumena where the things-in-themselves (ding-an-sich) exist, but our senses and observations can never really access them directly. The Feynman diagram lets us predict with enormous precision how particles will interact across space and time, but hides completely from view what these particles actually look like.

Another example of how math lets us ‘see’ the world we cannot directly access is the answer to the simple question: What does an electron actually look like?

Since the 1800’s, electricity increasingly runs our civilization, and electricity is merely a measure of the flow of electrons through space inside a wire. Each of us thinks of electrons as tiny, invisible spheres like microscopic marbles that roll through our wires wicked fast, but this is an example of where the human brain has created a cartoon version of reality based upon our ‘common sense’ ideas about microscopic particles of matter. In both physics and mathematics, which are based upon a variety of observations of how electrons behave, it is quite clear that electrons can be thought of as both localized particles and distributed waves that carry the two qualities we call mass and charge. They emit electric fields, but if you try to stuff their properties inside a tiny sphere, that sphere would explode instantly. So it really does not behave like an ordinary kind of particle at all. Also, electrons travel through space as matter waves and so cannot be localized into discrete sphere-like particles. This is seen in the famous Double Slit experiment where electrons produce distinct wave-like interference patterns.

So the bottom line is that we have two completely independent, mathematical ways of visualizing what an electron looks like, particles and matter waves, and each can facilitate highly accurate calculations about how electrons interact, but the two images (particle and wave – localized versus distributed in space) are incompatible with each other, and so we cannot form a single, consistent impression of what an electron looks like.

Next time we will have a look at  Einstein and his ideas about relativity, which completely revolutionized our common-sense understanding of space created by the brain over millions of years of evolution.

Check back here on Tuesday, December 13 for the next installment!

# Rules-of-thumb

There are at least two basic ways that we create associations. The first is associations in space. The second is associations in time.

Associations in space include recognizing static objects like chairs, trees, cars and people. The reason this works so well is that we live in a world filled with many different kinds of more-or-less fixed objects so that two or more people can agree they have similar attributes.

Associations in time include musical tunes and sounds, or associating one thing (cause) with another thing in the future (effect). For many of these dynamic associations like music, two people with normal hearing senses hear the same sequence of notes in time and can agree that what they heard was a portion of a familiar song, which they may independently be able to name if they have heard it before and made the appropriate associations in memory. But your exact associations related to the song will be different than mine because I associate songs with episodes in my life that you do not also share. Remember, the brain tags everything with patterns of associations unique to the individual.

The human brain is adept at pattern recognition. It can dissect its sensory information and see patterns in space and time that it can then associate with abstract categories such as a chair or a bird, and even specific sub-categories of these if it has been adequately trained (at school, or by reading a book on ornithology!). An upside-down chair seen in the remote distance is recognized as a chair no matter what its orientation in the visual field. A garbled song heard on an iPhone in a loud concert hall, or a particular conversation between two people in a noisy crowd, can also be detected as a pattern in time and recognized. The figure shows some of the brain connection pathways identified in the Human Connectome Project that help to interpret sensory data as patterns in space and time.

Patterns in space let us recognize the many different kinds of objects that fill our world. In the association cortex, once these identifications have been made, they are also sent on to the language centers where they are tagged with words that can be spoken or read. Once this step happens, two individuals can have a meaningful conversation about the world beyond their bodies that the senses can detect. Of course when both people say they have a specific category of objects called Siamese cats, they are most certainly associating that name with slightly different set of events and qualities corresponding to their cat’s personalities , fur patterns, etc..

The next step is even more interesting.

Just as the brain generalizes a collection of associations in space to define the concept of ‘cat’, it can detect patterns in time in the outside world and begin to see how one event leads to another as a rule-of-thumb or a law of nature. If I drop a stone off a tall cliff, it will fall downwards to the valley below. If the sun rises and sets today, it will do so again tomorrow. There are many such patterns of events in time that reoccur with such regularity that they form their own category-in-time much as ‘cat’ and ‘chair’ did in the space context. ‘If I visit a waterhole with lots of animals, there is a good chance that tigers or lions may also be present’. More recently, ‘If I stick my finger in an unprotected electrical outlet, I will probably be electrocuted!’. This perception of relationships is one of cause-and-effect. It has been studied by neurophysiologists, and is due to stimulation of part of the cerebellum and the right hippocampus. These brain regions are both involved with processing durations in time.

Over the centuries and millennia, the patterns in time we have been able to discern about the outside world have become so numerous  we have to write them down in books, and also put our children through longer and longer training periods to master them. This also tells us something very basic about our world.

Instead of being a random collection of events, our physical world contains a basic collection of rules that follow a ‘logical’ If A happens then B happens pattern in time. Physicists call these relationships ‘laws’ and their particular patterns in time and space can be discerned from measurements and observations made of phenomena in the world outside our brains. The brain can also work with these laws symbolically and logically, not by describing them through the usual language centers of the brain, but through a parallel set of centers that make us adept at mathematical reasoning.

In my next blog, I will discuss how mathematics and logic are intertwined and help us think symbolically about our world.

Check back here on Friday, December 9 for the next installment!

Space, Time, and Causality in the Human Brain
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4008651/

# A Stroke of Insight

Once the firehose of sensory information reaches the brain, a bewildering process of making sense of this data begins. The objective is to create an accurate internal model of the world that you can base your next decisions upon. To do this, all of the many bits of data flowing along the sensory neurons have to be knitted together somehow. Thanks to the unfortunate circumstances of minor strokes, brain researchers have been able to track down many of the important steps in this information processing.

You might have heard of the experiences of limb amputees who, for a time, experience the ‘phantom limb’ effect. The neurons having been severed still report back to the brain that their stimulation means the limb still exists, and for a period of time the amputee has to deal with the ghost limb that is not really there. In another bizarre situation, a stroke victim has a perfect understanding that their left arm belongs to them, but insists that their right arm belongs to a relative living 1000 miles away. This malady is called asomatognosia by neurophysiologists.

From many studies of how pinpoint strokes affect brain function, neuroanatomists have identified specific regions of the brain that allow us to integrate our sensory information and create a coherent model of the outside world as it exists in space and time. The first thing the brain has to do is to have a ‘sense’ of its own body and how it is located in space. It also has to identify this ‘self’ as being different from that of other people. If it cannot do this accurately, it cannot decide how to move in space, anticipate the consequences of that movement, or how to anticipate and empathize with the actions of other people. Nearly all of this activity seems to be relegated to a single area in the brain.

The temporoparietal junction (TPJ) takes information from the limbic system (emotional state) and the thalamus (memory) and combines it with information from the visual, hearing and internal body sensory systems to create an integrated internal model of where your body is located in space. The TPJ has left and right ‘lobes’ that control your ability to pay attention (right) and to anticipate other people’s emotions and desires (left). Patients with schizophrenia have abnormal levels of stimulation in the TPJ and cannot discern the intentions of other people. Stimulation of the right TPJ by placing electrodes in unesthetized patients leads to out-of-body experiences, schizophrenic behavior, and the phantom limb effect. The right TPJ tries to create a coherent body image from many different, and sometimes contradictory sensory inputs. When this process breaks down because the contradictory information is too strong to inhibit or ignore, you experience that you actually have two distinct bodies in space. This seems to be the direct, neural basis for out-of-body experiences.

But there is an even stranger brain region whose stimulation leads to an error in deciding where the body and self ends in space, and where the outside world begins.

The posterior cingulate body plays a huge role in self-location and body ownership. What this means is that we experience our body as having a definite location in space, and that this location is where you, the ‘Self’ is located. Strokes in this region cause asomatognosia patients not to recognize a limb as belonging to them. But you don’t have to be a stroke victim to experience this dislocation of body and self.

If you sit at a table facing a barrier that lets you see an artificial, life like right hand but not your real right hand, and you rhythmically stroke the real hand, eventually your brain gets fooled into believing that the artificial right hand is actually yours. If someone suddenly stabs the artificial hand, you will actually jump reflexively as though, for just an instant, the brain got confused about which was your real right hand being attacked!

The Posterior Superior Parietal Lobule gives us a sense of the boundary between our physical body and the rest of the world. When activity in this brain region is reduced, the individual seems to lose a sense of where their body ends and the rest of the world begins. The feeling is one of having ‘merged with the universe’ and your body is in some way infinite. Mindfulness practices such as meditation can modify the stimulation of this region and give the practitioner exactly this dramatic experience.

So you see, once sensory data gets to the brain, it is in for an amazing ride through many brain regions that help us build up the person or self that we feel we are through space and time.

By the way, for a fascinating introduction to these topics, read V.S. Ramachandran and Sandra Blakeslee’s book ‘Phantoms in the Brain: Probing the mysteries of the human mind’

Here is an interesting 2013 research paper in the journal Frontiers in Psychology ‘Alterations in the sense of time, space, and body in the mindfulness-trained brain: a neurophenomenologically-guided MEG study’ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3847819/

The connection between meditation and brain region function and stimulation is covered in this article: Mindfulness Practices and Meditation. https://neurowiki2012.wikispaces.com/Mindfulness+Practices+and+Meditation

But now let’s consider how the brain actually makes its models.

Check back here on Friday, December 2 for the next installment!