[math-fun] Elementary climate change explanations?
Bill Gosper: Warren, can you point me at an explanation how raising CO2 raises temps?
I think both sides agree that only ~10 meters of air is effectively opaque to the (IR) CO2 absorption frequencies, so the argument must be subtle.
--I don't agree, that far as I know is nonsense.
E.g., if the opacity claim is true, then why don't the CO2 notches in the spectrum http://i.stack.imgur.com/lfbu9.png clip at 0? Reradiation? Is the Nature article claiming that more CO2 will somehow deepen these notches?
--the usual explanation is, sunlight at temp=5800K falls to earth. The amount hitting earth is essentially unaffected by CO2. Then earth reradiates the same amount of energy as it received, but now using photons at 288K. Your spectrum picture shows sunlight is mainly "visible" say 500nm. Meanwhile the earth in trying to lose heat is mainly in the IR at say 9000 nm = 0.9 micron. (This is way off the right side of your picture, which hence for this purpose is pretty useless.) This attempt by the Earth to lose heat is frustrated by CO2 which is opaque to some of those photons. E.g. CO2 has big absorption bands at about 2, 3, and especially at 10-11 microns. As a result of this frustration, the earth in order to lose the same amount of heat as it receives, has to get a little hotter. Hence: more CO2 causes a "greenhouse" effect causing Earth surface to get a higher steady state temperature. This already was known by the late 1800s when CO2 and other absorption spectra were measured. You can (and I have) built a small greenhouse yourself, mine is made out of wire, sheet plastic, and a wood frame. It works pretty well to keep the inside warm. I forget but I think it keeps inside about 20F warmer than the outside on a sunny day. The reason is the same -- plastic and glass both are pretty transparent to visible but opaque to IR. The greenhouse effect is very important in determining planet temperatures. Venus has a huge atmosphere (about 90 times Earth's) containing mostly CO2 and water vapor, and as a result its surface is extremely hot, about 740K. If Venus had the same atmosphere as Earth, then its temp would only have been about 1.18 times Earth's and would be livable. The Earth's moon (with no atmosphere) has mean surface temp about 200K, well below Earth's 288K. If moon had same atmosphere as Earth its temp would be the same.
And why aren't those huge H2O notches surrounded by error fuzz, reflecting the variability of humidity? Is this variability only near the surface?
--H2O does cause a lot of noise. In fact the weather changes day to day, in case you did not notice. However over large times and large geographic areas, the noise averages out.
And how the heck many ppm was the "prehuman" level?
--the pre-human level of CO2 is known from ice core measurements (old air trapped in bubbles in old ice in antarctica and mountain glaciers). In fact here is a plot of CO2 level versus time going back to 400 Kyr ago from Vostok ice cores: http://cdiac.ornl.gov/trends/co2/graphics/vostok.co2.gif the present day level is about 400 ppm, but the Vostok levels plotted were 180-300. --also temperature vs time records are available too based on various "historic thermometers" e.g. based on isotope ratios. They track the CO2 levels pretty well http://en.wikipedia.org/wiki/Temperature_record
What is the optimal climate? If warming is bad, is cooling good? Or is all change bad? --I do not know these answers. But I think a change this large and this fast (+4C in 100 years), will be pretty devastating. It should yield large sea level rise and large ecological effects. We're adapted to the old earth. Re-adapting will be painful. Steven Sherwood (an author of latest sim study) was quoted in the Guardian "[the 4C rise] would likely be catastrophic rather than simply dangerous. It would make life difficult, if not impossible, in much of the tropics, and would guarantee the eventual melting of the Greenland ice sheet and some of the Antarctic ice sheet", with sea levels rising by many meters as a result.
The major thorn in the side of warming proponents is probably http://www.jpands.org/vol12no3/robinson600.pdf .
--Art Robinson is a nut. http://nelsnewday.wordpress.com/2013/09/14/oregons-own-wacko-art-robinson/ There's books about climate change deniers which prominently feature him and describe some of the weird aspects of his so-called career and funding sources. --I think the real thorns are (or should be) the questions of: What will climate change do? How much is its economic impact versus impact of climate protecting moves? I do not see many answers to those. Nor do I see much rationality from our so-called leaders, such as canceling USA breeder reactor research was a huge mistake. Not implementing large carbon taxes, is another huge mistake. These two mistakes have been ongoing for about 30+ years. If Art Robinson truly were the most important thorn in the side of "warming proponents" (whatever that meant), they would have already totally won. I do not think it is tenable to deny CO2-caused climate change. Too much evidence. It is, however, entirely feasible to influence the US congress to do nothing. One of the more important criticisms was the global temperature record over the last 150 years or so was called into question based on "urban heat island" effect, but a recent re-study by skeptic Richard Muller to examine this question concluded the dataset was fine and the compensation methods they'd been using, worked well.
________________________________ From: Warren D Smith <warren.wds@gmail.com> To: Bill Gosper <billgosper@gmail.com>; math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, January 1, 2014 10:46 AM Subject: [math-fun] Elementary climate change explanations?
Bill Gosper: Warren, can you point me at an explanation how raising CO2 raises temps?
I think both sides agree that only ~10 meters of air is effectively opaque to the (IR) CO2 absorption frequencies, so the argument must be subtle.
--I don't agree, that far as I know is nonsense.
E.g., if the opacity claim is true, then why don't the CO2 notches in the spectrum http://i.stack.imgur.com/lfbu9.png clip at 0? Reradiation? Is the Nature article claiming that more CO2 will somehow deepen these notches?
--the usual explanation is, sunlight at temp=5800K falls to earth. The amount hitting earth is essentially unaffected by CO2. Then earth reradiates the same amount of energy as it received, but now using photons at 288K. Your spectrum picture shows sunlight is mainly "visible" say 500nm. Meanwhile the earth in trying to lose heat is mainly in the IR at say 9000 nm = 0.9 micron. (This is way off the right side of your picture, which hence for this purpose is pretty useless.) This attempt by the Earth to lose heat is frustrated by CO2 which is opaque to some of those photons. E.g. CO2 has big absorption bands at about 2, 3, and especially at 10-11 microns. As a result of this frustration, the earth in order to lose the same amount of heat as it receives, has to get a little hotter.
Hence: more CO2 causes a "greenhouse" effect causing Earth surface to get a higher steady state temperature.
CO2 has a very strong absorption band around 15 μm from the bending mode. CO2 and H2O have complementary absorption bands, so their effects are additive.
This already was known by the late 1800s when CO2 and other absorption spectra were measured. You can (and I have) built a small greenhouse yourself, mine is made out of wire, sheet plastic, and a wood frame. It works pretty well to keep the inside warm. I forget but I think it keeps inside about 20F warmer than the outside on a sunny day. The reason is the same -- plastic and glass both are pretty transparent to visible but opaque to IR.
The greenhouse effect is very important in determining planet temperatures. Venus has a huge atmosphere (about 90 times Earth's) containing mostly CO2 and water vapor, and as a result its surface is extremely hot, about 740K.
There's no H2O left in the atmosphere of Venus. UV radiation breaks up the H2O, the hydrogen is lost into space, and the oxygen is too reactive to remain.
If Venus had the same atmosphere as Earth, then its temp would only have been about 1.18 times Earth's and would be livable. The Earth's moon (with no atmosphere) has mean surface temp about 200K, well below Earth's 288K. If moon had same atmosphere as Earth its temp would be the same.
And why aren't those huge H2O notches surrounded by error fuzz, reflecting the variability of humidity? Is this variability only near the surface?
--H2O does cause a lot of noise. In fact the weather changes day to day, in case you did not notice. However over large times and large geographic areas, the noise averages out.
And how the heck many ppm was the "prehuman" level?
--the pre-human level of CO2 is known from ice core measurements (old air trapped in bubbles in old ice in antarctica and mountain glaciers). In fact here is a plot of CO2 level versus time going back to 400 Kyr ago from Vostok ice cores: http://cdiac.ornl.gov/trends/co2/graphics/vostok.co2.gif the present day level is about 400 ppm, but the Vostok levels plotted were 180-300.
--also temperature vs time records are available too based on various "historic thermometers" e.g. based on isotope ratios. They track the CO2 levels pretty well http://en.wikipedia.org/wiki/Temperature_record
What is the optimal climate? If warming is bad, is cooling good? Or is all change bad? --I do not know these answers. But I think a change this large and this fast (+4C in 100 years), will be pretty devastating. It should yield large sea level rise and large ecological effects. We're adapted to the old earth. Re-adapting will be painful. Steven Sherwood (an author of latest sim study) was quoted in the Guardian "[the 4C rise] would likely be catastrophic rather than simply dangerous. It would make life difficult, if not impossible, in much of the tropics, and would guarantee the eventual melting of the Greenland ice sheet and some of the Antarctic ice sheet", with sea levels rising by many meters as a result.
Another very important effect of rising atmospheric CO2 is ocean acidification. Life cannot adapt as quickly as the CO2 is rising.
The major thorn in the side of warming proponents is probably http://www.jpands.org/vol12no3/robinson600.pdf .
--Art Robinson is a nut. http://nelsnewday.wordpress.com/2013/09/14/oregons-own-wacko-art-robinson/
There's books about climate change deniers which prominently feature him and describe some of the weird aspects of his so-called career and funding sources.
I must admit, I'm now disagreeing with Art concerning global warming. On the other hand, Art would like the country to build hundreds of new nuclear power plants. And for this, he gets my continued support. If we had not abandoned nuclear power, there would not be a global warming problem, at least none that could be blamed on the US. Let's not ignore the role of public education in creating a nation of idiots, believing the lies of both the oil companies and the antinuclear movement, and making the system self-perpetuating.
--I think the real thorns are (or should be) the questions of: What will climate change do? How much is its economic impact versus impact of climate protecting moves? I do not see many answers to those. Nor do I see much rationality from our so-called leaders, such as canceling USA breeder reactor research was a huge mistake. Not implementing large carbon taxes, is another huge mistake. These two mistakes have been ongoing for about 30+ years.
Carbon taxes without nuclear is pretty useless. Taxes won't reduce CO2 emissions, just raise the cost of energy. Carbon taxes will cause some shift to renewable, but the tradeoff is that windmills kill large birds, and solar farms destroy the deserts. Does anyone have any data on how long it takes to recover the energy invested in wind and solar facilities? Even convincing the public that there is a real problem will not get us closer to a solution. It will only shift the argument from whether climate change is human caused to how to mitigate. We will no nothing, just continue to burn carbon fuels.
If Art Robinson truly were the most important thorn in the side of "warming proponents" (whatever that meant), they would have already totally won. I do not think it is tenable to deny CO2-caused climate change. Too much evidence. It is, however, entirely feasible to influence the US congress to do nothing.
One of the more important criticisms was the global temperature record over the last 150 years or so was called into question based on "urban heat island" effect, but a recent re-study by skeptic Richard Muller to examine this question concluded the dataset was fine and the compensation methods they'd been using, worked well.
I'd like to know the specifics, or at least an example or two: What statements do you disagree with? Who said them? Which schools teach them? --Dan On 2014-01-01, at 11:34 AM, Eugene Salamin wrote:
Let's not ignore the role of public education in creating a nation of idiots, believing the lies of both the oil companies and the antinuclear movement, and making the system self-perpetuating.
Here's an actual example from my days in the NYC public schools some 60 years ago: We should donate 1% of our income to the United Nations. I hadn't learned the cuss words at that young age, but my reaction, keeping silent. was: "Why the f*** should I"? This wasn't just once, but year after year, like it was part of a script. -- Gene
________________________________ From: Dan Asimov <dasimov@earthlink.net> To: Eugene Salamin <gene_salamin@yahoo.com>; math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, January 1, 2014 12:34 PM Subject: Re: [math-fun] Elementary climate change explanations?
I'd like to know the specifics, or at least an example or two: What statements do you disagree with? Who said them? Which schools teach them?
--Dan
On 2014-01-01, at 11:34 AM, Eugene Salamin wrote:
Let's not ignore the role of public education in creating a nation of idiots, believing the lies of both the oil companies and the antinuclear movement, and making the system self-perpetuating.
On 1/1/2014 11:34 AM, Eugene Salamin wrote:
________________________________ From: Warren D Smith <warren.wds@gmail.com> To: Bill Gosper <billgosper@gmail.com>; math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, January 1, 2014 10:46 AM Subject: [math-fun] Elementary climate change explanations?
Bill Gosper: Warren, can you point me at an explanation how raising CO2 raises temps?
I think both sides agree that only ~10 meters of air is effectively opaque to the (IR) CO2 absorption frequencies, so the argument must be subtle. --I don't agree, that far as I know is nonsense.
E.g., if the opacity claim is true, then why don't the CO2 notches in the spectrum http://i.stack.imgur.com/lfbu9.png clip at 0? Reradiation? Is the Nature article claiming that more CO2 will somehow deepen these notches? --the usual explanation is, sunlight at temp=5800K falls to earth. The amount hitting earth is essentially unaffected by CO2. Then earth reradiates the same amount of energy as it received, but now using photons at 288K. Your spectrum picture shows sunlight is mainly "visible" say 500nm. Meanwhile the earth in trying to lose heat is mainly in the IR at say 9000 nm = 0.9 micron. (This is way off the right side of your picture, which hence for this purpose is pretty useless.) This attempt by the Earth to lose heat is frustrated by CO2 which is opaque to some of those photons. E.g. CO2 has big absorption bands at about 2, 3, and especially at 10-11 microns. As a result of this frustration, the earth in order to lose the same amount of heat as it receives, has to get a little hotter.
Hence: more CO2 causes a "greenhouse" effect causing Earth surface to get a higher steady state temperature. CO2 has a very strong absorption band around 15 μm from the bending mode. CO2 and H2O have complementary absorption bands, so their effects are additive.
This already was known by the late 1800s when CO2 and other absorption spectra were measured. You can (and I have) built a small greenhouse yourself, mine is made out of wire, sheet plastic, and a wood frame. It works pretty well to keep the inside warm. I forget but I think it keeps inside about 20F warmer than the outside on a sunny day. The reason is the same -- plastic and glass both are pretty transparent to visible but opaque to IR.
The greenhouse effect is very important in determining planet temperatures. Venus has a huge atmosphere (about 90 times Earth's) containing mostly CO2 and water vapor, and as a result its surface is extremely hot, about 740K. There's no H2O left in the atmosphere of Venus. UV radiation breaks up the H2O, the hydrogen is lost into space, and the oxygen is too reactive to remain.
If Venus had the same atmosphere as Earth, then its temp would only have been about 1.18 times Earth's and would be livable. The Earth's moon (with no atmosphere) has mean surface temp about 200K, well below Earth's 288K. If moon had same atmosphere as Earth its temp would be the same.
And why aren't those huge H2O notches surrounded by error fuzz, reflecting the variability of humidity? Is this variability only near the surface? --H2O does cause a lot of noise. In fact the weather changes day to day, in case you did not notice. However over large times and large geographic areas, the noise averages out.
And how the heck many ppm was the "prehuman" level? --the pre-human level of CO2 is known from ice core measurements (old air trapped in bubbles in old ice in antarctica and mountain glaciers). In fact here is a plot of CO2 level versus time going back to 400 Kyr ago from Vostok ice cores: http://cdiac.ornl.gov/trends/co2/graphics/vostok.co2.gif the present day level is about 400 ppm, but the Vostok levels plotted were 180-300.
--also temperature vs time records are available too based on various "historic thermometers" e.g. based on isotope ratios. They track the CO2 levels pretty well http://en.wikipedia.org/wiki/Temperature_record
What is the optimal climate? If warming is bad, is cooling good? Or is all change bad? --I do not know these answers. But I think a change this large and this fast (+4C in 100 years), will be pretty devastating. It should yield large sea level rise and large ecological effects. We're adapted to the old earth. Re-adapting will be painful. Steven Sherwood (an author of latest sim study) was quoted in the Guardian "[the 4C rise] would likely be catastrophic rather than simply dangerous. It would make life difficult, if not impossible, in much of the tropics, and would guarantee the eventual melting of the Greenland ice sheet and some of the Antarctic ice sheet", with sea levels rising by many meters as a result. Another very important effect of rising atmospheric CO2 is ocean acidification. Life cannot adapt as quickly as the CO2 is rising.
The major thorn in the side of warming proponents is probably http://www.jpands.org/vol12no3/robinson600.pdf . --Art Robinson is a nut. http://nelsnewday.wordpress.com/2013/09/14/oregons-own-wacko-art-robinson/
There's books about climate change deniers which prominently feature him and describe some of the weird aspects of his so-called career and funding sources. I must admit, I'm now disagreeing with Art concerning global warming. On the other hand, Art would like the country to build hundreds of new nuclear power plants. And for this, he gets my continued support. If we had not abandoned nuclear power, there would not be a global warming problem, at least none that could be blamed on the US.
That's pretty doubtful. And we haven't "abandoned" nuclear power; two new ones were approved in 2012. The most unfortunate, but understandable, decision was to develop uranium based power plants (for nuclear submarines) and abandon liquid thorium based power. The latter is far safer and produces only a tiny fraction of the waste. The only active pursuit of thorium breeders is in Norway, but I don't believe they're using liquid thorium salt.
Let's not ignore the role of public education in creating a nation of idiots, believing the lies of both the oil companies and the antinuclear movement, and making the system self-perpetuating.
--I think the real thorns are (or should be) the questions of: What will climate change do? How much is its economic impact versus impact of climate protecting moves? I do not see many answers to those. Nor do I see much rationality from our so-called leaders, such as canceling USA breeder reactor research was a huge mistake. Not implementing large carbon taxes, is another huge mistake. These two mistakes have been ongoing for about 30+ years. Carbon taxes without nuclear is pretty useless. Taxes won't reduce CO2 emissions, just raise the cost of energy. Carbon taxes will cause some shift to renewable, but the tradeoff is that windmills kill large birds, and solar farms destroy the deserts. Does anyone have any data on how long it takes to recover the energy invested in wind and solar facilities?
Last I read the EROI for wind was about 20 and for PV about 6. They're generally designed or a 30yr life, so it would be about 1.5yr an 5yr respectively. But of course the numbers are improving all the time, especially for PV where there's a lot of margin for improvement. Brent Meeker
Even convincing the public that there is a real problem will not get us closer to a solution. It will only shift the argument from whether climate change is human caused to how to mitigate. We will no nothing, just continue to burn carbon fuels.
If Art Robinson truly were the most important thorn in the side of "warming proponents" (whatever that meant), they would have already totally won. I do not think it is tenable to deny CO2-caused climate change. Too much evidence. It is, however, entirely feasible to influence the US congress to do nothing.
One of the more important criticisms was the global temperature record over the last 150 years or so was called into question based on "urban heat island" effect, but a recent re-study by skeptic Richard Muller to examine this question concluded the dataset was fine and the compensation methods they'd been using, worked well.
math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 1/1/2014 10:46 AM, Warren D Smith wrote:
Bill Gosper: Warren, can you point me at an explanation how raising CO2 raises temps?
I think both sides agree that only ~10 meters of air is effectively opaque to the (IR) CO2 absorption frequencies, so the argument must be subtle. --I don't agree, that far as I know is nonsense.
E.g., if the opacity claim is true, then why don't the CO2 notches in the spectrum http://i.stack.imgur.com/lfbu9.png clip at 0? Reradiation? Is the Nature article claiming that more CO2 will somehow deepen these notches? --the usual explanation is, sunlight at temp=5800K falls to earth. The amount hitting earth is essentially unaffected by CO2. Then earth reradiates the same amount of energy as it received, but now using photons at 288K. Your spectrum picture shows sunlight is mainly "visible" say 500nm. Meanwhile the earth in trying to lose heat is mainly in the IR at say 9000 nm = 0.9 micron. (This is way off the right side of your picture, which hence for this purpose is pretty useless.) This attempt by the Earth to lose heat is frustrated by CO2 which is opaque to some of those photons. E.g. CO2 has big absorption bands at about 2, 3, and especially at 10-11 microns. As a result of this frustration, the earth in order to lose the same amount of heat as it receives, has to get a little hotter.
Right. At the top of the atmosphere the temperature has to stay the same, so that the radiated energy equals the incident solar energy. But the CO2 causes more scattering of IR between the surface and the last scattering at the top of the atmosphere. So it's an IR insulator and the surface has to get hotter to still transfer the same power to space.
Hence: more CO2 causes a "greenhouse" effect causing Earth surface to get a higher steady state temperature.
This already was known by the late 1800s when CO2 and other absorption spectra were measured. You can (and I have) built a small greenhouse yourself, mine is made out of wire, sheet plastic, and a wood frame. It works pretty well to keep the inside warm. I forget but I think it keeps inside about 20F warmer than the outside on a sunny day. The reason is the same -- plastic and glass both are pretty transparent to visible but opaque to IR.
Your explanation of the global "greenhouse" effect is right. But the way actual greenhouses work has nothing to do with CO2 absorption. The just keep things warm by preventing convective cooling. Because glass and plastic are opaque to IR they don't let the energy into the greenhouse. Some guy even tested the theory by making a greenhouse out of panels of salt crystals that were transparent to IR. It got warm just like the glass ones.
The greenhouse effect is very important in determining planet temperatures. Venus has a huge atmosphere (about 90 times Earth's) containing mostly CO2 and water vapor, and as a result its surface is extremely hot, about 740K. If Venus had the same atmosphere as Earth, then its temp would only have been about 1.18 times Earth's and would be livable. The Earth's moon (with no atmosphere) has mean surface temp about 200K, well below Earth's 288K. If moon had same atmosphere as Earth its temp would be the same.
And why aren't those huge H2O notches surrounded by error fuzz, reflecting the variability of humidity? Is this variability only near the surface? --H2O does cause a lot of noise. In fact the weather changes day to day, in case you did not notice. However over large times and large geographic areas, the noise averages out.
Water vapor is the most effective greenhouse gas. It's absorption covers a lot more of the spectrum than does CO2 and there's a lot more of it. BUT it's only important as a feedback factor because it stays very close to equilibrium with the ocean surface. So it just serves to amplify the effect of CO2. In contrast CO2 will stay out of equilibrium for centuries.
And how the heck many ppm was the "prehuman" level? --the pre-human level of CO2 is known from ice core measurements (old air trapped in bubbles in old ice in antarctica and mountain glaciers). In fact here is a plot of CO2 level versus time going back to 400 Kyr ago from Vostok ice cores: http://cdiac.ornl.gov/trends/co2/graphics/vostok.co2.gif the present day level is about 400 ppm, but the Vostok levels plotted were 180-300.
Several hundred million years ago, CO2 levels were much higher; up to 5000ppm. But humans didn't exist then. If we, and the Earth's biota, had even a million years to adjust, doubling or tripling CO2 would not be a big problem. Brent Meeker
--also temperature vs time records are available too based on various "historic thermometers" e.g. based on isotope ratios. They track the CO2 levels pretty well http://en.wikipedia.org/wiki/Temperature_record
What is the optimal climate? If warming is bad, is cooling good? Or is all change bad? --I do not know these answers. But I think a change this large and this fast (+4C in 100 years), will be pretty devastating. It should yield large sea level rise and large ecological effects. We're adapted to the old earth. Re-adapting will be painful. Steven Sherwood (an author of latest sim study) was quoted in the Guardian "[the 4C rise] would likely be catastrophic rather than simply dangerous. It would make life difficult, if not impossible, in much of the tropics, and would guarantee the eventual melting of the Greenland ice sheet and some of the Antarctic ice sheet", with sea levels rising by many meters as a result.
The major thorn in the side of warming proponents is probably http://www.jpands.org/vol12no3/robinson600.pdf . --Art Robinson is a nut. http://nelsnewday.wordpress.com/2013/09/14/oregons-own-wacko-art-robinson/
There's books about climate change deniers which prominently feature him and describe some of the weird aspects of his so-called career and funding sources.
--I think the real thorns are (or should be) the questions of: What will climate change do? How much is its economic impact versus impact of climate protecting moves? I do not see many answers to those. Nor do I see much rationality from our so-called leaders, such as canceling USA breeder reactor research was a huge mistake. Not implementing large carbon taxes, is another huge mistake. These two mistakes have been ongoing for about 30+ years.
If Art Robinson truly were the most important thorn in the side of "warming proponents" (whatever that meant), they would have already totally won. I do not think it is tenable to deny CO2-caused climate change. Too much evidence. It is, however, entirely feasible to influence the US congress to do nothing.
One of the more important criticisms was the global temperature record over the last 150 years or so was called into question based on "urban heat island" effect, but a recent re-study by skeptic Richard Muller to examine this question concluded the dataset was fine and the compensation methods they'd been using, worked well.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Dan Asimov -
Eugene Salamin -
meekerdb -
Warren D Smith