Tuesday, November 22, 2022

Computer-aided artistry

 kw: experiments, reviews, art, dall-e, artificial intelligence

I can't draw, or paint, or sculpt. All the manual artistry in my family resides in my younger brother, who is a professor of art history. However, I can think creatively, which any child can do, as can many adults, if the skill hasn't been educated out of them. Perhaps if I had taken art lessons, long ago…

But now, AI can help! I read a review of three "art generator" applications and decided to try them. Rather than write a blow-by-blow account, I'll present my progress and conclusions. The three applications are 

  • DALL-E 2, the first and currently the best.
  • MidJourney, which runs on a server within the Discord environment.
  • Stable Diffusion, also in Discord, but a beta version of an improved product is in DreamStudio.

I planned to use three prompts. A prompt is a text instruction, which can be up to the size of a tweet. The longer and more detailed, the better the results (usually). Two of the three applications return four small images; you can pick one to edit. The editing tools depend on the application. Quite frankly, the tools in MJ and SD are obscure and apparently limited (more experienced folks can correct me on this; just comment). The tools in DE2 were easier for me to comprehend and to use, so I did a lot more with that one.

I used only one prompt with MJ and SD, and then gave up. The first prompt:

Mountainous Landscape in the style of the Hudson School

DALL-E:

I selected the second one to edit. I first downloaded the image, a 1024x1024 pixel PNG file nearly 2 Mby in size. Converting to a JPG reduced it to 475 Kby. This panel is shown about half size. I'll discuss the little bit of editing I did below.

MidJourney:

I selected the first one to (try to) edit. Saving the 1024x1024 image produced a 662 Kby WEBP file. Converting to a JPG reduced it to 517 Kby. This panel is shown full size.

I couldn't figure out whether extending the image (I'll get to that shortly) is even possible, so at this point I stopped and didn't send MJ any more prompts.







Stable Diffusion:

This single return seemed odd. It must be two halves of a panorama. Saving it produced a 1024x1024 PNG file (though I had to select the size at the outset; a slider runs from 512 to 1024). 

I couldn't do much with this. I tried a different prompt, which produced slightly better results, but not too pleasing to me.









I returned to DALL-E 2. I primarily wanted to extend the image. I'm interested in images big enough to use for screen saver wallpaper (1920x1080). In the edit toolset, you extend an image by adding a Generator Frame (AKA a Marquee), moving it where you like, and clicking Generate.

I first put the marquee next to the image and clicked Generate. The marquee filled with a different take on the prompt, independent of the image next to it. I clicked Reject, moved the marquee to overlap about 1/8th of the image and clicked Generate. Much better. I clicked Accept. This is the image I kept, 1920x1024 in size (a bit panoramic, but I like that). I reproduce it here half size.


You might have noticed that all DALL-E images have a little color bar at lower right. This is to be used if you want to print an image on canvas, to guide the color-rendering software; you'd hide the lower edge behind the frame. To make an image for use on the screen one must extend it enough to crop that lower edge out for the finished image.

The editor lets you erase part of an image and paste in other stuff. You can also start by downloading an image and editing it, rather than using a prompt.

I used the second and third prompts at this point. I'll show the final, extended images (half size).

Second prompt:

Still Life with apples, pomegranates, and bananas in the style of Paul Cezanne


This image is 1984x1280. The original 1024x1024 extends from the middle of the blue ewer to the end of the top banana, and to a little below the edge of the table.

Third Prompt:

Futuristic city on hilly alien planet with violet sky and two moons


This panoramic image is 2624x1024; the original square is about the middle 40%. The added moons showed up during extension.

DALL-E 2 gives a new user 50 free credits for the first month of use, and 15 free credits per month thereafter (use 'em or lose 'em). Each time you click Generate you consume one credit. Credits can be purchased, $15 for 115. Using this program can consume a lot of time, and some funds. It can easily become addicting. I have read of online artists who produce images and sell them as NFT's. That's one way to fund the addiction!

At this point I had 35 credits left. I couldn't resist trying one more prompt:

Desert landscape with mesas and saguaro cactus

The 4 returns:

These look like scenes near Tucson. I chose #3 and extended it, to both sides and below:


This image is 2624x1664, plenty big enough for me to use as wallpaper. It consumed 8 credits, leaving me with 27. Yo-ho! What to do next? Stay tuned!

Friday, November 18, 2022

The real cycle(s) of life

 kw: book reviews, nonfiction, biochemistry, krebs cycle, citric acid cycle

Sorry, Lion King, the circle of life is actually something hidden from most of us. There are powerful wheels that turn inside every cell in our bodies (and in all bodies, from bacteria right on up). The most important of these has a name that strikes fear in students of biochemistry: the Krebs cycle. Hans Krebs received (with Fritz Lipmann) the Nobel Prize in 1953 for his elucidation of the citric acid cycle, which is usually called by his name.

This illustration, from Transformer: The Deep Chemistry of Life and Death by biochemist Nick Lane, shows how the cycle produces ATP, the energy carrier used by all living beings (on Earth at least). The steps after ATP is produced (lower left) regenerate molecules used for the next go-round. Citric acid (in the form of citrate when in solution) is the master of ceremonies.

For those interested in the structural chemistry shown here, the black balls represent carbon, the small gray balls are hydrogen, and other atoms such as oxygen or sulfur are represented by open circles with a letter inside. The dashed lines with a - sign attached to all the COO groups indicate that the electronegativity is not localized to either of the oxygens.

This book is a loving biography of the Krebs cycle and related biochemistry. Dr. Lane does his best to explain the reactions within it, and in the reverse cycle, and the environment in which this chemistry is active: on both sides of the membranes of mitochondria. These cycles are the machinery that runs our cells.

A common understanding of the Krebs cycles has what we call the forward cycle transforming energy into a form we can use (ATP), and the reverse cycle gathering energy from the environment, such as the electrons pumped by photosynthesis, to make energetic molecules, mainly sugars. But there is a whole lot more to it than that.

Other crucial parts of these metabolic cycles include "red protein" or ferredoxin, which catalyzes reactions that run too slowly otherwise, forcing electrons onto many of those COO- groups seen in the diagram, or onto intermediate chemicals that pass them onward to COO-. Another is shown as CoA here; it is "coenzyme A", a somewhat larger chain of atoms and small rings, including some phosphate groups and amino (nitrogen-hydrogen) groups. The body makes CoA from vitamin B5 (pantothenic acid). Every cell on Earth needs it to function.

It may surprise those who haven't studied biochemistry that so very many of the chemicals of life are acids. In popular culture, "acids" are powerfully corrosive chemicals such as sulfuric acid (battery acid) and hydrochloric acid (muriatic acid or swimming-pool acid). We may know that the sour taste of vinegar is due to acetic acid, and that lemons are sour because of citric acid. So not all acids are that fierce! Surprisingly, every cell in your body is powered by a cycle that begins and ends with citric acid. It's about a whole lot more than oranges and lemons! The COO- group in an organic molecule is called "carboxylate", and when the minus sign is satisfied by an attached hydrogen, the COOH group makes the molecule a carboxylic acid. These are not corrosive. Rather, they are necessary for metabolism to function. Also, a protein is a long chain of amino acids, and all amino acids contain the compound group NH2-COOH. We are built of acids!

How are these molecules built? Their building blocks are produced by tools created in the Krebs cycle. This cycle has so many uses, it has to be regulated so as to avoid conflict between producing ATP for energy and producing molecules for body construction. Dr. Lane likens a cell to a city, with lots of activity going on. At the core of all the processes to run the city are motors, and the motors all have the same brand name: Krebs. A motor is a good metaphor, for as energy flows through a motor, the motor spins. In one quote from biophysicist Harold Morowitz, "Energy flows, matter cycles."

Once the opening few chapters have described the Krebs cycle in sufficient detail, and provided examples of how the "motor" runs, later chapters delve into the question, "Which came first, genetics or metabolism?" A fundamental fact about all the reactions in the cycle are that they are reversible. We learn in early Chemistry classes that when you have a reversible reaction, it can be driven either way by changing the concentration of other chemicals in its environment (typically by adding one of the products or reactants to a solution in a beaker).

Recent experiments—usually meaning in the last 5-10 years—have shown that these reactions can proceed most of the way around the cycle with very little "driving". Having a metallic or metal-oxide substrate for the acids to temporarily attach to also seems to facilitate matters. Even more recent experiments have shown that pressure and heat—here meaning pressures of several tens of atmospheres to several hundred atmospheres and heat around boiling or not much above boiling—facilitate these reactions chains, and if certain products are removed, they are like conveyor belts or assembly lines to produce the kinds of molecules that are necessary for life. This has led to the hypothesis that life began at hydrothermal vents in the ocean deeps.

It occurred to me as I read this that hydrothermal vents should have been much more active a few billion years ago than they are today. When I was taking geochemistry, I remarked to the professor one day that radiogenic heating (heating caused by elements such as uranium breaking down) must have been six times greater than now, four billion years ago. This would lead to much faster plate tectonics (He was surprised; he had never thought of that). At that time the deep sea "black smokers" and other hydrothermal features had not yet been discovered. Eons ago submarine vents would have been correspondingly greater in extent and activity. It seems the immediate post-Hadean era could have been ideal for biologic life to begin.

Nearer the end the book turns from life and life's origins to disease and death. It is no surprise that, if our mitochondria age and wear out, the cycling of metabolism is affected. I have been wondering for a long time, if our mitochondria age, and their DNA accumulates SNP's even faster than the DNA of our cells (nuclear DNA), how do babies get born with brand-new mitochondria? Is there a corrective mechanism? There is! And it is not quite described, but briefly outlined on pages 140-141. The author calls it a "clean-up operation in the female germline" to prepare the half-million of mitochondria that fill each oocyte (egg cell).

Also, mitochondria, and the metabolic cycling that spins endlessly around their membranes, are implicated in cancer. This metabolic cancer origin hypothesis is not yet well known, and is controversial where it is known. But it makes more sense than the chain of oncogene disruptions posited by the earlier hypothesis (and it is no more than a hypothesis, not a theory).

Further, dying and death are metabolic in origin. I can't say I grasped the entirety of Dr. Lane's description, but it made sense as I read it. We really are a lot like "The Wonderful One-Hoss Shay" of Holmes's poem, that was constructed to have no weakest part. If nothing external goes wrong, we may live to great age, and then rather abruptly suffer general organ failure, when everything seems to go wrong at once: "He died in his sleep." My great uncle, having outlived his wife by a few years, at the age of 102 was working a field on his farm. He stopped the tractor and walked to one of the hired hands to say, "I feel a little tired. I'll take a little nap." Inside, he lay down and passed away peacefully. I can't think of a better way to go.

The epilogue is titled "Self". It reconsiders the question, "Which is primary, genetics or metabolism?" The conclusion (stated at the outset and then supported by evidence): "Genes never supplanted the deep chemistry of cells. They conserved it, and they built on it." The primary difference between the Krebs cycle of four billion years ago, and now, is the cluster of enzymes (built by genes) that catalyze the reactions, facilitating energy flows hundreds of times more rapid. Otherwise animal life would not be possible, and plant life couldn't have produced bamboo that is able to grow a few feet per day.

The book made me wistful. I started college as a chemistry major because I wanted to become a biochemist. Three changes of major later, I graduated as a geologist. I had a great career, but I miss chemistry. This book has become my new favorite for the year.

Thursday, November 17, 2022

Where are the Millenarians?

 kw: longevity, multiverse, musings

Among those who are so unwilling to die that they will try anything to circumvent the inevitable, I find a strange bunch who pin their hopes on the Multiverse.

The reasoning goes like this: the Multiverse has uncountable numbers of alternate universes that differ from the one we inhabit in numerous ways, from negligible to minor to rather major. They say (this is an approximate quote), "When someone in this universe is faced with death, so are many 'copies' in similar universes. Suppose you die in this universe, and so do many of your 'copies', but some of your 'copies' don't die in their universes. Can it be that your consciousness somehow traverses between universes, so that you find yourself in one of those where you didn't die? This can happen again and again. Therefore, nobody really dies, they just get a transfer to a place where they didn't die."

Let us consider for the moment that this supposition is true, and one may, just before (or during) dying, transfer to another universe where life goes on. If this can happen again and again, can it keep happening for a long, long time? If "nobody really dies," why is it that our universe doesn't seem to have anyone in it who has hung on for hundreds of years?

This couple recently celebrated their 81st anniversary. They are 98 and 102. Shouldn't there be someone out there celebrating anniversary #100, 200, or 1,000?

How is it that our universe isn't at the receiving end of lots of transfers? Further, if you get a transfer to the universe next door, what happens to the consciousness of your 'copy'?

Where are the thousand-year-old people?

Saturday, November 12, 2022

Salt and Vinegar explored

 kw: experiments, vinegar, salt, acidity, ph, photo essays

Kitchen chemistry: putting vinegar on an oxidized penny doesn't seem to do anything. Adding table salt makes the penny shine right up. Also, I get lime deposits in the ceramic cup holder in the bathroom. I've tried the same method. Adding salt to vinegar on the lime makes it much easier to clean off.

Question: Does the salt make the vinegar more acidic?

I have some pH paper that I bought when I was a chemistry major many, many moons ago. Here are a couple of pieces on a salad plate, initially dry, near the paper dispenser with its scale. The pieces have a pH near 5 because of carbon dioxide in the air, which makes any moisture in the air shift from a neutral pH of 7 to about 5.6, the natural pH of rainfall.

The color bars on the scale are 1, 3, 5, 7, 9, and 11.

I next added a couple of drops of vinegar to each piece of paper. They became a little more red, indicating a pH near 4. Then I put some salt on the lower one. These pictures show what happened (not much!):


The salt is visible in the pic on the right, on the lower paper. If I try hard, that piece of pH paper may look slightly redder than the other. But really, there is no measurable change in the pH after adding salt.

So why does vinegar with salt added clean pennies, and help remove lime scale more rapidly?

I think the effect is due to kinetics. Adding table salt, sodium chloride, causes an equilibrium reaction such that some of the acetate ions in the vinegar solution shift their "allegiance" to sodium ions, and some of the hydrogen ions are then free to "work with" chloride ions. Hydrogen chloride, or hydrochloric acid, is a stronger acid than the acetic acid in vinegar. "Stronger" doesn't mean it has a different pH necessarily. It refers to how strong the reaction is when it encounters a material it can attack, such as copper patina or lime scale.

Without doing experiments for which I don't have the equipment, I can't go further. The hypothesis, "Hydrogen chloride attacks susceptible materials more effectively than hydrogen acetate, in low-pH solution" will do for now.

Tuesday, November 08, 2022

Senses - a Baker's Dozen

 kw: book reviews, nonfiction, natural history, science, senses, physiology

I discovered something shocking about Chinese soup. At a church potluck dinner we all enjoyed a bowl of soup from a big pot one Chinese sister brought. It had an intriguing taste, similar to soups made with Star Anise, but subtly different. In my bowl I found a black pod. I was told it was the seed pod of water chestnut. I bit into it, and had quite a surprise! Suddenly the soup tasted awful. Water tasted like battery acid. Every taste was distorted, in quite unfortunate ways, for about a day. When I told this to the cook, she laughed and said, "You aren't supposed to eat Chinese spices!" I couldn't stand to eat or drink anything until the next day.

I found out that this seed pod goes by many names, including Devil's Pod. You can get them from two sources: Chinese food stores, and Etsy, where they are sold for use in crafts (not as a food item).

The words "taste" and "flavor" have different meanings. For a physiologist, "flavor" includes both taste and smell. That is but one tidbit I find in Sentient: How Animals Illuminate the Wonder of Our Human Senses by Jackie Higgins. Another is that the five kinds of taste we know may actually number seven. In addition to sweet, sour, salty, bitter, and savory ("umami", triggered by glutamine from protein), some researchers have found hints that our tongues have sensors for calcium (a different kind of salty) and fat.

We all learned that the "five senses" are sight, hearing, taste, smell, and touch. These are the senses that have visible organs. However, we also have senses of balance, hunger, and a host of others that may number more than 20. Indeed, our eyes sense two different regimes of light. When the only light in a room is a candle, but we can still see colors in all except the gloomiest corners, our sight is near the threshold of photopic vision. Light bright enough for us to distinguish colors is sensed by the color-selective cones in our retina. Moonlight, particularly when the moon is a few days past full, or a few days shy of full, is sensed by the color-blind (actually blue-green sensitive) rods in our retina, and we can see rather well by scotopic vision.

There is actually another network of light-sensing cells in our retina that connect to our body clock, so that day/night cycles reset it daily. Long-term experiments with people in caves and bunkers have shown that without this daily resetting, our body clock tends to run on a 25-hour cycle. Thus, in addition to color vision and night vision, our eyes have a third function related to our sense of time. Body temperature, blood pressure, and alertness run in daily cycles; the "afternoon sleepies" aren't just because of that big lunch you ate.

The 12 chapters in Sentient each focus on a different sense. Eleven of these are known in humans: 

  • color vision
  • night vision
  • hearing
  • touch (but this is a multi-modal group of senses with different receptors for each)
  • pleasure and pain (two levels of stimulation of one set of receptors)
  • taste
  • smell (by itself, or as bundled into "flavor" with taste)
  • desire (pheromones)
  • balance
  • time
  • proprioception (required to touch your nose with eyes closed)

The twelfth (Chapter 11) is direction, which we'll get to shortly. There is a bonus chapter on a 13th sense, the electric sense in the bill of a platypus. Sharks and other aquatic predators also have it. Humans don't; getting an electric shock stimulates nerves directly, but there is no sensor that allows us to know the subtle electric fields around moving animals. That's probably a good thing. If we had an electric sense it would be overloaded by the pervasive 60- (or 50-) cycle hum found everywhere except unpopulated areas and Amish homes, and by the multitude of electromagnetic signals that bring our favorite programs to AM and FM radios and TV antennas.

The iconic animal in each chapter is renowned for excellence, or exceedance, in the chosen sense. For example, the Mantis Shrimp of Chapter 1 has, not just the three color sensors that we (and most primates) have, but a total of TWELVE, including one or two that see ultraviolet. Apparently, in the extremely colorful environment of a coral reef, it is thus better able to discriminate specific prey by their colors. Chapter 6 on Taste tells of a large catfish. Catfish are known to have numerous taste receptors all over their bodies; one researcher calls a catfish a "swimming tongue". The Goliath Catfish is the largest; think of a ten-foot tongue swimming around.

I was quite interested in the directional sense that some people exhibit (Chapter 11). Many animals including migrating birds are found to have pieces or chains of magnetite crystals that let them sense, and perhaps even see, the magnetic field of the earth. When migrating, they follow the direction of the force, or go at an angle to it; at the end of the trip, they may sense the steepening angle as they approach the magnetic pole, and finish their journey when the angle reaches a certain degree. Do humans have a magnetic sense? Many experiments include quite a number that seem to say "Yes", but some seem to say "No" and others are equivocal. If humans do not have a magnetic sense, that'll be odd, because so many mammals do have one, and members of nearly every other group of animals have it. We do have something, or some of us at least, for there are those who always know which way is north, or home, or another chosen direction, even after being taken somewhere blindfold.

This is so far my favorite book this year: Fascinating, packed with very interesting information, and easy to read.

I'd like to end with speculation about a fourth sense that may be located in our eyes. It is something I've noticed after I turn out the lights in my bedroom. In the dark, with my eyes closed, I seem to be able to see my hand and arm move, and also the blanket, when I rearrange the covers. This is not scotopic vision somehow seeing through my eyelids. The rods are sensitive to blue-green light peaking at 500 nm (normal green-sensitive cones peak about 540 nm). Rods cut off on the longwave side at about 600 nm, but hardly any light shorter than 600 nm gets through our red-colored eyelids, because of the blood in them.

By my bedside is a clock radio with bluish LED numerals. I have a pink filter over it so it doesn't light up the whole room (the radio is poorly designed). It is plenty bright enough for me to see the blue color. I can't see it at all through my eyelids, even when fully dark adapted. But there is a little reddish light in the room from a couple of pilot lights on equipment, and a pinkish light that comes through the windows from skyglow. When there is still enough light in the room to see faint colors it looks like this (picture edited to look like what I remember):


On moonless nights, after my eyes have become dark-adapted, it looks more like this:


The slight bluish hue is typical of scotopic vision. A scene by moonlight looks bluish, even though the moon's actual color is brown. Although rods outnumber cones 20-to-1 (120 million rods and 6 million cones), they are ganged together for greater sensitivity, which greatly reduces the sharpness of the scene.

Now, here is what I noticed about a year ago. I typically turn out the light around 11 or a little later. Even before I am fully dark-adapted, when my eyes are closed and I readjust the covers or move my hand, I can see the movement of my hand and the covers, and if—still with closed eyes—I look around the room, I can see the outlines of what is in the room, a little more faintly than if I open my eyes, but distinctly. The windows seem the brightest. Further editing of the photo from above yields this, which matches what I see:


It is perhaps twice as blurry as scotopic vision, and the light and dark areas are a little different. If I hold up my hand and wave it about it seems black. If I raise my head and look at the clock radio, with its dim blue numerals, I don't see them at all.

Is it possible that I am seeing right through my eyelids? If so, it must be by sensors that see by the pink light from the windows, and/or the faint red light of the pilot lights. It could be something else entirely.

I have considered that my brain may be conjuring a dim memory of the room, as part of a normal vision function that anticipates what is "normal", priming the visual system to detect any differences. My brain knows where my hand is; it knows how the bedcovers move when I shift them; it knows where the windows and furniture are. It could know enough to meld proprioception (knowing where my head is pointed and where my hands are) with this calculated scene so that the scene "stays put" when I turn my head.

I cannot decide at present whether I am actually seeing, or constructing, what I sense.

Tuesday, November 01, 2022

Fresh water's best expression

 kw: book reviews, nonfiction, limnology, lakes, natural history

This is a picture of me walking (well, standing) on water. It's easy when it's frozen! This was taken at Pactola Lake in the Black Hills of South Dakota during my graduate school days. In the background waterfowl are taking advantage of the unfrozen part of the lake. Pactola is not a natural lake but a reservoir, a combination of water supply for Rapid City and flood control for Rapid Creek. The lake's area is about 740 acres (300 hectares), which makes it a middle-sized lake. While it is several miles long, it is at most just over half a mile wide, which makes it perfect for canoeing. That's something some friends and I did a few times.

According to John Richard Saylor, in Lakes: Their Birth, Life, and Death, there are about one-and-a-quarter million lakes on Earth, of 10 hectares (24.7 acres) or greater extent. Some limnologists count lakes larger than 1 hectare, saying there are some 8.5 million.

Right away I am going to fault the author for utterly ignoring metric units. I suspect many readers will wonder, "What's special about 24.7 acres?" A simple foot note or parenthetical note could clear that up. Of course, 10 hectares is arbitrary, but nearly everywhere except America it's at least understandable. There isn't a well-agreed-upon way to distinguish a lake from a pond, so a line has to be drawn somewhere.

Limnology is the study of fresh water in all its forms, although glaciology has its own niche when studying frozen water. However, this book says little about rivers, except as feeders or drains of lakes. The book follows a simple classification scheme. Glaciers produce the most lakes by far, either by gouging out basins or by depositing moraines. 

A few years before moving to South Dakota I spent a few weeks doing geology in an area called 20 Lake Basin, above Yosemite in the Sierras. This image from Google Earth has 15 labels, but two of them are "...Lakes", and there are several lakes visible with no label. Most of them are gouged-out lakes. The basin is surrounded by glaciers. I swam in nearly every one of the lakes shown.

Landslides sometimes form lakes by blocking a stream. Such lakes seldom last long, but there are a few that have persisted for centuries. Then there's Quake Lake near Yellowstone in Montana, formed by an earthquake in 1959. It's about six miles long, and seems to be here for the long haul. Time will tell.

I don't know if people make more dams than earthquakes and landslides do, but there are thousands of artificial dams and their attendant ponds or lakes. Some are huge, such as Lake Powell (half dried up at present) on the Colorado River: 25-40 miles wide, with an area around 160,000 acres (65,000 hectares).

There is a kind of allure about damming up a stream. As a child, I was like many of my friends, in that we sometimes dammed up a narrow spot in a local creek, to see how high we could get the water to rise. It was typically no more than an inch or two. I spent a few otherwise idle hours moving stones and sand into place to patch up first one, then another, "escape route" the water would take as it rose.

On the Wikipedia page for Lake Powell, it is noted that sediment with a volume of 11 billion gallons settles in the lake every year. Since 1969, capacity has been reduced 7%. This is part of the ordinary life cycle of any lake. Once formed, it will be gradually filled with sediment. During its "active lake" phase, a lake will host wildlife that varies according to the local climate. 

All lakes eventually fill up, or are drained in some way (often by human activity in recent centuries). Given time, a lake that avoids being drained turns into a marsh, then a bog, then a meadow, and in time, it may leave little trace. 

The author points out a few lakes that are millions of years old (Lake Baikal in Russia comes to mind). He does a quick study with us about what it takes to get a lake to last so long, or longer. While some areas of Earth are up to a few billions of years old, the erosion cycles they have been through would have erased any lakes that formed in that time. It's likely that a tenth of a billion years is about the limit.

It's an enjoyable excursion into the subject. Much recommended!