Tuesday, March 6, 2018

VR sword haptics, part 2

Another option for a VR haptics sword is to use jets mounted to the tip of the sword create reaction torques. In spacecraft design, these are called cold gas thrusters. They work on the same principle as any jet engine or rocket - mass ejected from a system causes an equal and opposite momentum change in the system - except there is no combustion. A minimum of three nozzles (or maybe just one plus a direction actuator) mounted equally spaced in a plane normal to the sword would be able to generate torques about any combination of two axes.

What kind of pressures and flow rates would be required to generate the required thrusts? For an ideal nozzle, the imparted thrust equals the mass flow rate of the jet times the velocity of the jet:

F = ṁv

For supersonic flow from a diverging nozzle, jet exit velocity and mass flow rate are dependent on the inlet pressure, the nozzle throat diameter, and the type of gas. Hot, low density gases have better performance, but compressed air at room temperature is certainly easiest to use for this application. For a 200psi (14atm) inlet pressure, a 4mm diameter nozzle throat will allow the flow rate required to produce about 20N of thrust, which at a 1m lever arm of course means about 20N*m felt at the handle.

What would a system designed to generate these flows and pressures look like? Ideally, very little mass would be attached to the sword, especially the tip, so that swinging it around doesn’t take too much effort, and so that the jets have less momentum to counter. A proportional control (throttleable) air valve rated to the required pressure and flow rate costs maybe a few hundred dollars and weighs maybe 0.5kg, if well designed. A few of these would be too heavy to place at the tip, but a pipe running from handle to tip would be able to conduct the air with minimal pressure drop and keep the jet response time on the order of 20ms.

A tank large enough to hold enough pressurized air for even a minute of gameplay would be too large and heavy to be mounted to the handle, so the sword would have to be connected to an external supply hose. For continuous operation, a large stationary compressor and tank would be required - think large 6ft (2m) tall machine shop setups.

So, like the gyroscope approach, a reaction jet haptic sword would probably be too expensive for the consumer market. However, a reaction jet haptic sword would be significantly simpler to design and prototype, though it has the disadvantages of being tethered and very loud.

Animal Suffering

I’ve never given vegetarianism or veganism serious consideration. Nor seriously engaged with an adherent of either about their motivations. It seemed commonplace enough to be unremarkable - a respectable preference that wasn’t for me. What little thought I’d given to the idea could be boiled down to the argument from the natural order - many animals are carnivores, so why should we restrict ourselves? While exploring the effective altruism movement last year, I came across the first chapter of Singer’s Animal Liberation [1] where he lays out the core of the philosophical argument for minimizing animal suffering and was recently intrigued enough to read the rest of the book and engage fully with the ideas.

The core of the argument asks us to acknowledge that, just as there is no difference between the human races when it comes to being human, there is no difference between humans and other animals when it comes to the capacity to suffer. Just as being human entitles one’s interests to equal consideration among all humans, so having the capacity to suffer entitles a being’s interest in not suffering to equal consideration among beings. “Pain felt by an animal should be given the same weight as the same amount of pain felt by a human.” If we acknowledge this, then we cannot morally cause that pain in an animal for the purpose of a mere preference in taste.

This argument took some effort to internalize, but I now accept its validity. The text addressed well the various conceivable misunderstandings or objections - for example, equal consideration does not imply equal treatment or rights, plants do not suffer in the same way that we do, modern factory farm conditions indeed inflict considerable pain - as well as my naive rationalization from nature. Unlike the carnivores, humans have the capacity to think morally. With our intelligence comes the responsibility to consider such ethical questions as minimizing suffering.

“I hope that anyone who has read this far will recognize the moral necessity of refusing to buy or eat the flesh or other products of animals who have been reared in modern factory farm conditions. This is the clearest case of all, the absolute minimum that anyone with the capacity to look beyond considerations of narrow self-interest should be able to accept.” [1]

Singer doesn’t attempt to make any strong arguments about the ethics of pain-free killing because killing animals is more ethically ambiguous than inflicting suffering. For me, it comes down to a question about the difference between human and animal self-awareness. What is the meaning of life for a cow? Does it have goals beyond the next meal or meaningful relationships? I like this thought experiment: is there a difference between three short pain-free cow lives and one pain-free cow life three times as long? “In the absence of some form of mental continuity, it is not easy to explain why the loss to the animal killed is not … made good by the creation of a new animal who will lead an equally pleasant life” [1]. I currently tend to agree, but not without considerable uncertainty. Although Singer warns that “if we are prepared to take the life of another being merely in order to satisfy our taste for a particular type of food, then that being is no more than a means to our end” [1], he also “can respect conscientious people who take care to eat only meat that comes from [animals with pain-free lives].”

The problem is that achieving this practically is difficult. Even “humanely certified” dairies (separation at birth, bull calves sold to slaughter, continuous pregnancy and milking) or “free-range” egg farms (aggressive laying schedules and feeding, beak trimming, male chicks sold to slaughter) or “painless” slaughterhouses (crowded feedlots, mistakes with the captive bolt gun) are not without significant suffering [2]. The pressures of the market encourage treating the animals as machines, especially when the consumer is uninterested in the details.

I find it is important to keep in mind the utilitarian equivalence of even seemingly minor suffering to pain we feel as humans. We can imagine the physically unpleasant moments of being farmed for flesh and wonder if we would inflict that on ourselves for the pleasure of eating the animal product.

Flesh and animal products truly free from inflicted pain might be ten to twenty times more costly to raise than factory farmed flesh. Very small herds, caring husbandry, uneconomical resource utilization, careful killing. An expensive delicacy. At what point is the taste worth the effort? Veganism is the obviously unambiguous moral choice, but I do not think I have the willpower for that just yet.

Because pain-free products aren’t readily available for sale, I must rely on some kind of metric to help me make choices to reduce the suffering I sponsor. Sufficient strides have been made at the vanguard of humane farming of animals that some products are available for which the embodied suffering is tied mostly to processes surrounding slaughter. For example, pasture-raised hens can have fairly pleasant lives until the hours before slaughter where they are crated up, shipped, racked upside down, and conveyored off to stunning. For this reason, animals that produce a lot of product over their lifetime, like dairy cows, rank very high (better) on a calorie per suffering metric [3]. If we assume equal suffering per death and assign equal value to the suffering of different animals, salmon and chicken flesh are roughly equivalent in calorie per suffering, eggs rank ~10x higher (better) than that, beef ranks ~10x higher than eggs, and dairy ranks ~20x higher than beef. Based on this metric, if one must consume animal products, one should should stick with dairy. And if flesh must be occasionally tasted, then beef is least harmful.

However, there is another metric I consider important - greenhouse gas emissions. Climate change is an immediate crisis that will cause great suffering to beings and damage to ecosystems on a scale much larger than animal farming (irrecoverable diversity loss, many more beings affected by 2-4 orders of magnitude [4]). Though harder to quantitatively compare, minimizing its impact by reducing greenhouse gas emissions is just as important as reducing farm suffering, if not much more. Studies of the relative environmental impacts of animal farming show that beef is ~10x worse than all other animal products [5]. This penalty knocks beef down to rough equivalence with eggs in the ranking above. 

My diet has changed considerably over the last ten years. From what was approximately the average American proportions, I estimate that my flesh consumption has decreased by 5-10x. Dairy and eggs have probably stayed about the same. Vegetable consumption has increased maybe 2-4x. These changes have been motivated solely by exposure to new foods, increased consideration of my physiological health, and consciousness of the impact that my choices have on greenhouse gas emissions. But now, the acknowledgement of the equivalence of the capacity for suffering among beings demands further change.

As I rarely eat chicken or beef already, it is easy for me to eliminate terrestrial flesh entirely. Eggs and fish consumption is next for reduction. Finally, I plan to slowly work on reducing dairy intake. This is possibly my brain forecasting and generating loopholes, but I think it is important to remember that because being perfect often distracts from other valuable things, drastically reducing is often almost as good as entirely eliminating.

Trying out this new perspective feels disconcerting - a disturbance in the norms I’ve been comfortable with. I catch glimpses of what people might have felt when deciding to treat slaves as humans. It helps me appreciate the struggle this idea continues to have in gaining credence in our culture. The core of the idea also seems to be hidden behind so many contrasting and confusing prescriptions, that the effort to examine it and form a considered opinion is a deterrent. It seems especially important to lay out clear cases and engage in debate.

I’d love to hear and discuss other perspectives and rationales.

[1] Singer. Animal Liberation. 1975. Chapter 1

Individual Real Estate Investment

I find myself confused at the temptation people have to try buying real estate as an investment. Typically, their plan is to do a bit of research into “hot” areas or engage with a property manager in order to select a property that they imagine has some special potential for either appreciating in value or generating large rents. Then they plan to buy the property, hold and maintain it for some time while collecting rent, and finally sell it for some total gain they expect to be better than if they had invested their money elsewhere. I don’t understand the attraction. And I’m confident the expected returns are poor relative to other investments. I see buying individual investment properties as analogous to buying individual equities on the stock market.

It is uncontroversial that stock picking is for suckers. Putting bets on what will win is fun and exciting, as I am told gambling is. It indulges the human brain’s overconfident capability for predicting the world. But it doesn’t pay off. The vast majority of actively managed funds are outperformed by broad market indices [1]. That is, an investment in a total stock market index fund almost always generates greater returns than careful picks by the investment industry’s top experts. An amateur trying to pick individual winners based on what she read in a few smart sounding articles cannot expect to perform better.

From an expected value perspective, an individual picking a particular real estate property is just as misguided as an individual picking a particular stock in order to “beat the market”.

An index fund provides an inexpensive, diversified investment option. It is designed to eliminate the high risk of picking individual stocks while still capturing the returns from the growth of a market as a whole. The real estate market has investment products designed for the same purpose - real estate investment trust (REIT) indices. A total market REIT index captures the returns from the growth of the real estate market as a whole. It is designed to eliminate the high risk of investing in individual properties.

If index funds are shown to be the best long-term holding of stocks, then a total market REIT is the best long-term holding of real estate. The only reason someone would gamble on individual properties is because they mistakenly expect to beat the total market index.

I spent some time trying to find a study on the real estate market similar to the report above on the stock market. A study that investigates whether or not the vast majority of real estate portfolios containing individual properties are outperformed by a total market REIT index. But either it hasn’t been investigated or I couldn’t find it. So we will have to settle for the argument from structural similarity.

A total stock market index fund returns the average performance of all the companies in the market, weighted by market capitalization. It follows that, over some period, there is a 50% probability that a random selection of stocks will outperform the index fund, and a 50% probability that a random selection will underperform. Despite this prior probability, greater than 90% of managed funds (selections of stocks chosen not randomly, but by intelligence) underperform the index fund. This is due to adjustments to the portfolios that hurt the fund (i.e. picking a promising replacement stock that then drops in value) as well as fees and transaction costs.

A total market REIT index returns the weighted average performance of all the real estate investment companies in the market. Just as in the previous case, a random selection of real estate properties has at best a 50% probability of outperforming the total market index. Equivalently, a real estate investor starts with a 50-50 chance of underperforming an index. Combined with the lesson from the stock market, where the attempt to try to land on the outperforming side of that 50-50 ends up doing the opposite, it appears that investing in individual real estate properties is unwise, as the investor can choose to take the guaranteed average through a REIT and walk away.

As an aside, I was curious to look at historical data. This paper [2] finds that global long-term (1870-2015) mean returns from real estate (including both capital gains and rents) are similar to returns from equities (stocks). The paper also breaks down the results by nation and “medium-term” - for the USA over the last 70 and 40 years, total market returns to stocks have been a couple percentage points higher than those to real estate. But this is only tangentially relevant here.

An objection might be raised that because a total market REIT invests only in real estate investment companies on the public market, it does not capture returns on the housing market as a whole. While this is true, the returns on a total market residential REIT should be comparable to returns on housing [2].

The most common objection to this argument for placing individual stocks and properties in the same risk/reward class is that the motivated investor can discover some type of special information about a property (ostensibly because it is a tangible thing that can be plainly inspected and evaluated) that is much more difficult or disguised in the case of a stock. Perhaps they have noticed a trend in housing prices in the area, or an office park is about to be constructed nearby, or they heard there may be a treasure chest buried in the yard.

The implicit assumption is that the situation is comprehensible. That the investor has a sufficiently accurate model of the adjacent world - of all the major factors that might significantly affect the value of the property - that she can invest with some confidence. But this is classic overconfidence effect in action - an underestimation of the complexity, the randomness, the noise in the world. Maybe that upwards trend was actually an overvalued bubble, or maybe the increased traffic around the office park makes the neighborhood less desirable for families. The same cognitive biases that cause stock pickers to fail apply to real estate investors.

Full time fund managers have access to as much, if not more, publicly available information about the companies they evaluate as a real estate investor has about economic and social forces that govern housing values. There is no such thing as a simplified market playground or some kind of home court local advantage.

Even in areas with booming markets, there is still large local variability. Say an area happens to experience growth rate that exceeds the market average by 2% (something that roughly represents what the SF Bay Area has achieved over the last 25 years). With a relatively large standard deviation in annual property value changes of 9% (typical for the area) [3], only about 58% of properties beat the market average and 42% underperform, assuming a gaussian distribution in values. This same standard deviation means a 16% chance of underperforming the market by greater than 7%. Underperforming the market index doesn’t necessarily mean that the property is losing value, just that an investor could be seeing greater returns if she had taken the easier path of investing in a total market REIT index.

What about the advantages of leverage that are easily accessible in real estate through a mortgage? The potential returns on a leveraged investment are indeed higher, but the potential losses are equally large. Leverage merely magnifies the position - both risk and reward. It doesn’t affect the conclusions above. And besides, the companies held by a REIT hold plenty of leveraged positions on their books already, so there is nothing special about being able to hold an individual leveraged position.

I’ve been assuming the goal of an investment is to maximize returns. The other reason that someone might choose to gamble on individual properties or stocks is because it is fun. Index funds are boring. Some people enjoy the effort of research and the thrill of a bet. But it should be clear that they are spending money for that enjoyment. Just like I spend lots of money on my paragliding hobby.

[1] S&P Global. SPIVA® U.S. Scorecard. 2017.

Thursday, March 30, 2017


A nice interactive map displaying data from HOLC "security maps" of the 1930s.


It should not be surprising, but it is illuminating to read such blatant discrimination on seemingly mundane government reports.

"DETRIMENTAL INFLUENCES: Infiltration of colored residents. There are now about twelve families scattered over the area indicated"

"Infiltration of undesirables/dark Portuguese/Oriental store-keepers/Orientals: serious threat/subversive races exist"

Monday, February 6, 2017

VR sword haptics

The latest virtual reality craze is great because it looks like we are finally getting some nice, relatively affordable visual displays along with integrated controller systems. I played ping pong on one and was impressed. I think that playing VR ping pong would improve someone’s ability to play ping pong IRL. That’s a test of usefulness.

But the big problem with VR is haptics. A VR system can only achieve so much by simulating vision and sound (and sometimes proprioception). To really step up the level of immersion and to open up new interaction possibilities, incorporating the sense of touch is required.

Laura, John, and I were discussing this and thought that sword haptics could be very compelling. Sword haptics have a surprisingly simple set of basic requirements: to create torques in two axes. Any hit on a sword, above the handle, could be simulated by a torque in the handle. The exception is a direct thrust down the axis of the sword, which could easily be fudged into a torque in gameplay.

It is somewhat counterintuitive to accept that only torque is required because when we imagine the case of two swords slamming together and locking (think of the two fighters staring at each other across the swords and trembling with exertion), we think of them pushing forward with their swords, not swinging. But because the point of contact is removed from the center of pressure of the wrist(s), the handle sees only a torque. The fighter’s body can only apply as much force forward as the wrists can maintain in torque with a lever arm to the point of contact. As a rough spec for testing feasibility, two human wrists on a handle can apply a torque of about 30N*m statically.

One way of implementing haptics like this is to use cables anchored to various points in a room to apply forces to the tip of a mock sword. We tested this out by building a prototype system that allowed us to move around the cable winch anchor positions in order to find a workable configuration. In order to keep up with someone swinging a ~1m long sword around, the cable reel-in/reel-out linear speed must be high (about 7m/s), but not unreasonably so. The winch system to drive the cable could cost <$100.

The problem with this approach is the tradeoff between usable space (the volume where forces on the sword tip can be controlled in all directions), and the cables interfering with the movement of the sword wielder. The usable space is defined by a convex hull with vertices at each of the anchor points, and with shallow concavities in each of the faces (because the cables must apply an exponentially increasing tension in order to achieve an outward force as the tip approaches the surfaces of the hull). There is a decent workable solution without interference if the wielder stands at or just inside the boundary of the hull, facing into it with the sword. But this limits the wielder’s flexibility to turn around or to swing the sword too far in either direction around her. On the other hand, if the player stands inside the hull, in order to have greater usable space and flexibility, the cables become even more likely to interfere. Bummer. A more complex potential solution might have the anchor points moving around so as not to interfere with the player.

Another idea for implementing a haptic sword is to use an array of gyroscopes mounted to a sword handle. The gyroscopes would be double gimbal mounted to the handle so that the handle would be able to pivot in any direction freely while the angular momentum vectors of the gyroscopes remain pointed in the same direction. In order to simulate a sword hit, both axes of each of the gimbals would be locked, thus coupling the angular momentum of the gyros to the handle. The momentum of the gyros combined with the swing of the sword would cause a reaction torque, stopping the rotation of the sword abruptly. Three gyroscopes, each mounted orthogonally to the others, would be required to ensure a reaction torque regardless of the angle of the sword or direction of swing.

Such a sword could have interesting applications, like in fencing game where blades easily deflect, and touch for only a tiny fraction of a second. However, in this unactuated implementation, the wielder of the sword would still be able to translate the sword handle without resistance. In other words, after the gyroscopes lock and stop the rotation of the sword stroke, the wielder could still push (translate) their sword right through their imaginary opponent’s sword. It wouldn’t be a very realistic feeling.

For a more realistic feeling, the gyroscopes would have to be actuated to apply a sustained torque to overpower the wrists and move the sword handle. This actuated implementation is slightly trickier to think about. The gyroscopes would effectively act as CMGs (control moment gyroscopes). Changing the direction of a gyroscope’s angular momentum by rotating it with an actuating motor causes a reaction torque about an orthogonal direction. Reaction torque equals the gyro’s angular momentum cross the angular velocity at which the gyroscope is rotated.

𝛕 = L x Ω

The entire gyroscope array would still be mounted inside one double axis gimbal. To simulate a sword impact, the gimbal axes would be locked, and then, to apply further reaction, the actuating motors would rotate a combination of the gyroscopes in the array to create the appropriate reaction torque in the handle.

Gyroscopes are magical things, but they have limitations. For one, as a CMG is torqued by its actuating motor, its angular momentum vector changes, which in turn changes the direction of the reaction torque vector. In order to maintain an output torque about an invariant axis (so that the sword hit seems like it is coming from one constant direction), two gyroscopes can be paired together in a complementary arrangement called a scissored pair. The scissored pair cancels angular momentum in the unwanted direction and results in a sinusoidal torque output in one direction. This means that in addition to varying the input angular velocity to adjust the strength of the simulated sword hit, the torquer motors must also vary their speed to maintain constant output torque. To ensure a reaction torque could be applied regardless of the angle of the sword, three orthogonal scissored pairs would be required, for a total of six gyroscopes mounted inside the gimbal to the handle. (It might be possible to generate reaction torques in all directions without scissored pairs, that is, with just three orthogonal gyros, but the scissored pair arrangement is easier to think about, simpler to control, and possibly more efficient.)

The other limitation of gyroscopes arises from the law of conservation of (angular) momentum. A gyroscope cannot be made to apply a torque about the axis along which its angular momentum is aligned. As a CMG is torqued, it gradually aligns with the direction of output torque as it transfers angular momentum to the system. When it becomes aligned, it has reached what is called saturation - it can’t transfer any more momentum in that direction. What this means for the sword is that the gyroscopes can’t sustain a simulated hit infinitely. Flight simulators have this same problem - they can’t simulate an infinite acceleration. Besides increasing the stored angular momentum, the way to get around this is to reset the angular momentum direction slowly, under a sensory threshold.

So what does this gimballed triple scissored pair system look like? It is big and heavy. If we want the sword handle to be able to apply 30N*m for a period of 1s, it would weigh around 7kg (a gallon of milk weighs 4kg) and take up a space of about 0.3m in diameter, probably very optimistically. Not exactly something that would be feasible to swing around. These numbers are based on my rough calculations assuming carbon fiber rotors spinning at about 30,000rpm, geared brushless DC torquer motors, slip rings for the gimbals, and very nice expensive design. It would be a fun optimization problem though.

What if we reduce the torque spec by about a fifth, to 6N*m? The rotors and torquer motors and gear train get somewhat smaller in size, so the diameter of the system could be reduced by about half. But the complexity of the system is still the same, so the system mass is would only be reduced by about half as well. Switching to a system with three individual gyroscopes (instead of scissored pairs) also doesn’t help much for system size and mass because the rotors have to get bigger.

So it’s not an obviously economically or practically feasible design for a consumer VR system, but it could certainly be prototyped with some money and effort.

While investigating this idea, I evaluated many other combinations of gyroscopes and gimbals to achieve the desired effects, but all ultimately had problems complexity. Reaction wheels can’t generate a high enough torque for an extended period, single gimballed scissored pairs aren’t able to decouple their momentum from the handle, and while an array of single gimballed gyroscopes potentially can, that arrangement is more complicated than a single encircling double gimbal. Gyroscopes are fun to think about!

So this idea of realistic sword haptics gets tossed for now.

hmmm, what about reaction jets???

Monday, December 19, 2016


A family member was under the misguided impression that gerrymandering was not much an issue these days. I collected some images to show just how alive gerrymandering is today.

Explained graphically:

The tactics are called cracking and packing.

North Carolina congressional districts in real life:
Outcome:  10 republican (77%), 3 (33%) democratic congresspeople. Vote count: 53%R, 47%D. Another way of calculating this difference is with a concept called the efficiency gap.

It's also easy to see in the results how the packed 1st, 4th, and 12th districts are consistently won by democrats by much higher margins (~35%) than the republican districts (~17%).

Close up of Charlotte:
Look at that and try to say racial gerrymandering doesn't happen in the US these days.

Observe what happens in district shapes around 1995.

This happens all over the USA:

NC 4th district:
Texas 35th district:

Thursday, December 8, 2016

renewable energy advocates are not misguided

A friend was under the impression that the DAPL protesters and others fighting fossil fuels are misguided in their efforts. He believed that people are being misled about renewable energy sources like solar and wind, and that the full nuclear option is the only rational choice for stopping global warming.

I hoped to broaden his perspective.

First, as an example case, I imagined what the reservation would look like powered by only solar:

"The electricity demands of the people on Standing Rock Reservation could be easily met with 2km^2 of solar power. That is about 1/5000 of the land area of the reservation.

Let’s do the math:

The average electricity consumption per capita in the USA in 2013 was about 1800W. That is way high for a rural reservation without industry, but we’ll use it. The population of the reservation is about 8000. So we have 15MW. Solar panels can roughly provide about 10W/m^2 of continuous power, accounting for solar intensity variations, nighttime, and efficiency. In order to get 15MW, we therefore need 1.5km^2 of solar. The area of the reservation is about 10,000km^2. So the solar plant would take up about 1/5000th of the land area of the reservation.

The population density of the entire Dakotas is about 4x higher, but we haven’t considered wind (which has a similar power density to solar), and it’s really fuckin windy in the Dakotas.

What about their gas guzzling cars? Liquid fuel is an issue, but if they all drove electric the reservation’s electricity consumption would increase by about 50% (transportation energy consumption is similar in size to residential consumption, but EVs are way more efficient)."

Oil production is not really related to electricity production so we kinda had two conversations going.

"Among the many reasons for fighting these oil pipelines, the most important is to force the costs of oil production as high as possible in order to encourage transition over to sources of energy that do not release CO2. They are fighting the right battle.

I do agree that, in addition to renewables, we should be seriously ramping up the construction of nuclear plants, not tearing them down. Their relative environmental impact is indeed minimal.

Excellent technical discussion about sustainable energy, and my main reference."

He was a little confused on units, so I elaborated.

"Thanks for recommending to look at the NREL maps to check my quick estimate. However, it seems I come to an even better conclusion! I am looking at this map:

Perhaps you are confusing the units of kWh/m^2/day with W/m^2? I also read off a value of about 5 for annual averaged solar resource at standing rock, but it is in units of kWh/m^2/day, not W/m^2. The two units are both units of power/area, but they have different quantities (like the difference between an erg and a joule). If you do the conversion (I like to do it by hand, and then check it with something like WolframAlpha), it turns out that 5kWh/m^2/day equals about 200W/m^2. (I prefer converting to W/m^2 because it lets me compare to consumption easily.) You can also double check this by looking at a weather station which often records insolation. For example

Squish down that yellow hump over 24 hours and you get about 100W/m^2, cuz it’s winter.

Then, we do our PV efficiency cut of 15%, which leaves us with 30W/m^2. Looks like I was being pretty conservative before with my estimate of 10W/m^2.

The fine print for that map says that it already takes into account all the qualifications you raise:
- it assumes fixed, flat plate collectors,
- it takes into account the number of sunny days and the seasons,
- it takes into account sun angle, and
- it even takes into account nighttime because it is using units of energy produced per day (24 hours). Thank you, NREL for making this easy!

Now we’re down at 0.5km^2 of solar for all of standing rock!

A quick glance at another NREL study suggests this all is totally reasonable. Already, ND could generate about 25% of its household electricity by installing PV on the roofs of all its small buildings.

Now, the problem of intermittency. I did not explicitly state in my last comment that I was assuming energy storage coupled to the solar PV. I am. This is pretty standard, and becoming more so (think molten salt storage, Tesla Powerwalls, and electric vehicles). For example, the Crescent Dunes Solar Thermal Plant, which generates about 0.1GW, has about 1GWh of storage. That means that even when it is pitch dark, assuming the battery is charged, it can put out 0.1GW for 10 hours. Plenty to cover daily swings in consumption.

What about winter/summer swings? Our friends at NREL suggest seasonal resource ranges from 3 to 7 kWh/m^2/day. That’s a factor of 2, and usage maybe doubles, so that puts our estimate for the reservation back at 2km^2."

Here's what 2km^2 looks like in the context of the reservation (the red rectangle):

Finally, my friend was misinterpreting an out of context quote from David MacKay, british physicist turned sustainable energy expert. He thought that MacKay was unequivocally stating that we should ditch all solar and wind effort in favor of nuclear. My response:

"I believe you are misunderstanding MacKay’s intention in that excellent video. He is answering a question about what he thinks is the ideal optimum scenario for a carbon neutral UK. He says all nuclear. I think you, MacKay, and I are all in agreement on the promise of nuclear as a steady, clean power source that could be scaled up to meet all our electricity needs. However, the world does not work in optimum solutions. It works by patch jobs. It works through diversity and compromise. Fighting exclusively for the optimum solution has always been the downfall of well meaning engineers everywhere. Especially in the current cultural climate where nuclear is unjustly frowned upon and in the current economic climate where the LCOEs for renewables are just about the same for fossil fuel or nuclear, solar and wind are strong palatable options for making progress against climate change. There is no excuse for holding out for an optimum solution when progress can be made immediately.

Secondly, while his point is correct that when you already have enough nukes to last through the winter, building more solar or wind is a waste, we are nowhere near that point yet. At this point, during the transition, the existing fossil fuel plants will act as the backups.

Thirdly, MacKay repeatedly states that his energy portfolio recommendation would be different for other countries. The UK is significantly worse off compared to the USA in terms of solar and wind resources.

My point through all this (besides destroying your absurd original claim that “their reservation would literally be one giant solar panel” :) ) is to emphasize just how important a role solar (and wind) will play (and are already playing) in transitioning away from fossil fuel electricity. Obviously the Dakotas will not be the first place we start to see proliferation of solar PV (but installed wind in ND is 10th nationally), but sometime soon, it will be economical, particularly if we keep fearing nukes. Do you think the >50% of Americans who oppose nuclear will be able to be convinced to install 100s of nuclear plants instead of solar or wind in the next 10-20 years? I wish. Perhaps we can override their protests and force them (hi, mr trump!)? Perhaps with an insane public education effort. But those don’t usually work :/