Sex and Math: The Zero to Two Percent of Our Actual Lives That Rules Our Lives

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Someone (probably Louis CK) pointed out that even though we all obsess about sex (and being attractive to the persons with whom we long to have sex), it’s actually a very, very tiny part of one’s actual (as opposed to our wishful) life.
[Of course, any species that survives needs to have a strong drive for procreation, or they won’t leave enough offspring behind to carry on. (Just ask pandas what happens when they seemingly couldn’t care less… They are, if you hadn’t noticed, nearly extinct in the wild. Rats, rabbits, ants and cockroaches survive by being unbelievably fertile — and sneaky.]
Let’s try to do some math on this.
Have you ever tried figuring out what percent of your entire life has consisted of you actually making love (having sex) with another person?
I am not counting masturbation here because
  • I don’t trust any data on this, even Masters and Johnson, so any guesses on my part would be just that
  • I’m only talking about having sex with another person, either Penis-In-Vagina or any other sort of sexual activity, which I am not about to list here —  use your own imagination if you want to. 
  • And no, I’m not going to tell you anything about my own habits or those of anybody else I know. However, if you want to tally up your OWN time doing ‘that’, feel free.
 
Let’s look at the high end of the spectrum of those having lots of sex first.
My guess is, based on observations of personal experience and what I’ve observed with people I knew well:  that only with the very horniest newly-weds or with a couple who have just entered a super-sensual, brand-new sexual relationship,  time would a couple be spending, say, as much as four hours a day actually’doing it’. Why? For one thing, sex is exhausting. Also, the tissues involved are delicate and can only take so much rubbing, no matter how well lubricated they might be. Plus the couple need to sleep, eat, wash, and most likely do something productive like going to school or work.
That very high figure for someone getting a HUGE amount of nookie is 4 hours  out of 24 hours in a day, or 1/6, which is about 17% of the time. Let me repeat: that’s extraordinarily high, and from my own observations (thin walls allow one to hear… and so on) only seems to last for the first few weeks, because then they get sore, exhausted, sated, and somewhat jaded.
After that, they’d be lucky to be ‘doing it’ for an hour or two a day, which is between about 4 and 8% of the time.
But let’s compare that to the rest of their lives for this remarkably horny and lucky couple… Let’s suppose that they are 20 years old, and let’s suppose that they each had sex a few times in high school and after (college, military, working, whatever, and I’m making no assumptions about whom they are doing this with). Maybe they got laid 2 – 3 times a week from age 16, at about an hour each time (girls, feel free to scoff at my suggestion that a typical male teenager can actually last that long) plus, now that they are not living at parents’ home any more, they have had several tumultuous, sexy relationships one after the next, each time spending an average of 2 hours actively ‘doing it’ each 24-hour period, for the last two full years.
So let’s make that say 6 hours a week for 2 years plus 28 hours a week for the last 2 years – and these people are those who are towards the very far right hand end of the frequency curve distribution. I don’t know whether the distribution of ‘nookie-hours’ if graphed, is ‘normal’ or ‘skewed’one way or the other. But in any case, most people ‘get’ a lot less sex than this. In fact, when I was in high school and college, the vast majority of my (male and female) friends my age or younger were virgins. No, I’m not going to tell you what age or to whom I lost my virginity. I will keep my fond memories to myself, and hope you will do likewise.
Here’s the arithmetic:
Denominator (total hours lived) for this very lucky pair of 20-year olds: 24 hours per day time 365. days per year times 20 years old equals 175,200 hours total that they have lived so far. (D)
 
Numerator ( total hours having sex with another person since they were born) Assuming this very sexually active couple who have sex 6 hours a week times 104 weeks (two years from ages 16 – 18) plus 28 hours a week times 104 weeks (two more years, aged 18 – 20) equals 3,536 hours having sex, grand total, (N)
 
I used a calculator to divide N by D and I get about .02, or two percent of their life so far. That’s tops.Two percent of their life.
I suppose it is theoretically possible that a married couple of 60 years, at age 75, has been actively having sex a full hour every single day of their married life (since age 15). Extremely rare, I know… But for this tiny handful of  extraordinarily horny, happy, healthy and lucky people, I get a grand total of 1/30 of their life, or a tad more than 3%. And I bet that you could probably hold a nice party for every single one of these lucky 75-year-old living couples — from all over the world — in the cafeteria of the elementary school nearest my house. Not sure how much standing room there would be, but this is out of our current world population of over 6 billion people. So they are outliers of outliers of outliers.
The vast majority of people would have much, much less sex than that, I predict with confidence.
Additionally: a good number of people die before losing their virginity. I have no idea what percentage do, but during the old days, it was probably well over half  of them, infant mortality rates being what they were — and I’m assuming that most of those babies and children who died at very early ages from disease, murder or accidents weren’t being sexually abused during their short, sad lives… Many soldiers die in combat having had sex perhaps 0 to 6 times in their entire lives, if I can trust various memoirs I’ve read by veterans of various wars…
So that’s the far left hand end of the ‘bell-shaped’ curve: 0%.
While there are a tiny handful or two-to-three percenters out there, I would bet (if we could test it somehow) that over 99.99% of the population has sex, on average, between ZERO and TWO percent of our entire lives.
But, oh, how we obsess over ‘it’!!

When was the last time you spent a night under the stars?

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If it’s been a while since you spent time looking up at the heavens with your naked eyes, binoculars and telescopes, looking at planets, stars and galaxies, then this Saturday might be your night.

The Hopewell Observatory is having an open house on Saturday, July 2, 2016, and we have a variety of scopes to look through. Some of the scopes will be under our roll-off roof and some will be rolled out onto the small lawn outside the observatory itself.

Mars, Jupiter and Saturn will be very conveniently placed for viewing right at sundown, and if it’s dry and clear enough, we should be able to see the Milky Way. Many nebulae, open and globular clusters, galaxies, and double or triple stars will be visible as well.

You are invited!  And it’s free!

The location is about an hour due west of Washington DC by way of I-66, near the town of Haymarket, VA. For detailed directions, follow this link, which I posted for one of the dates which got canceled because of bad weather. Ignore the date, but do pay attention to the fact that we have no running water! We have bottled water and a composting toilet and hand sanitizer. Plus makings for coffee, tea, and hot chocolate – all gratis.

picture of hopewell

The picture above is of one of our telescope mounts, which carries several telescopes and was set up to take astrophotographs at the time. Below is a picture of the outside of the observatory shortly after a snowstorm.. Notice that there is no dome – instead, the galvanized steel roof rolls back on the rails and columns to the right of the picture when the scopes are in use.

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If you have your own telescope, feel free to bring it. If it needs electricity, we have an outdoor 120VAC outlet, but you should bring your own extension cord and plug strip.  If you want to stay all night, that will be fine, too! If you feel like bringing a cot or a tarpaulin and a sleeping bag, that’s equally OK by us! Show up at or near sunset, and stay until the sun comes up, if you like!

Warning: the area definitely has insects, such as ticks and chiggers, which appear to avoid everybody else and to do their best to attack me. I strongly recommend long pants, shoes/boots, and socks that you can tuck the pants into. Tuck your shirt into your pants as well, and use bug spray, too. I have personally seen plenty of deer, cicadas, moths, wild turkeys, squirrels, and birds, and I have heard from a neighbor that a bear tried to eat his chickens, but other than the insect pests, the wildlife stays out of your way.

Again – for detailed directions, look at this link.

Any Moth Experts Out There?

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We found these two beautiful moths that flew into the operations cabin at the Hopewell Observatory a couple of nights ago, and we have no idea what type they are. Never seen them before and can’t find any images identical to them. (One species is similar, though.)

Any suggestions will be welcome.

Ain’t they purty li’l things?

And when they opened their wings they were even more spectacular, but I didn’t get a good shot.BTW the yellow-and=red moth is sitting on the struts of a telescope made by Alan Bromborski.

Movie about Ramanujan

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I recommend going to see ‘The Man Who Knew Infinity’, about the famous Indian mathematician Srinivasa Ramanujan, whose originality astonished the greatest mathematicians of his day, G H Hardy and Littlewood.

It’s a historical romance rather than a documentary, so there are parts that are completely made up — like German Zeppelins bombing Cambridge University during World War 1, and off-duty British soldiers beating up Ramanujan for being a foreign draft dodger. A weird aspect is that the actors and directors don’t seem to be on the same page about the correct pronunciation of Ramanujan’s name. They mostly put the accent on the third syllable, where as far as I can tell it should be more like ruh – MAHN – uh – jahn.

However, there was such a bombing near-by during the war, and the British army did have temporary hospitals for wounded soldiers during that war, and Ramanujan did contract tuberculosis and die at the sadly early age of 32, after he had gone back home. And apparently his mother had indeed been intercepting letters to and from his wife, so they were completely cut off from each other, as the movie states.

And of course, they can’t really explain much of the mathematics – heck, I can’t understand much of anything that Ramanujan did! Even his simplest formulas are way, way over my head! Dramatizing partitions of a number was probably a wise move, since it’s one of the few things that an ordinary person might understand.

 

 

 

 

Evolution is SLOW

A long time ago I had a course by Prof Jim Sandeful of Georgetown U and Dr Monica Neagoy on teaching with “discrete math” — very useful and interesting stuff that often does not get discussed in the standard American curriculum. I enjoyed it a lot.

Among other topics, I decided to write a little computer program that would model exponential random decay of radioactive elements. (Iirc I did this in Pascal and in BASIC, on the C-64, IBM-PC, Apple II, and Commodore Amiga. That was fun.)

One subtopic that came up, but which I never figured out how to model, was how to describe the frequency of some trait (eg red hair, striped tail, or growing a third eye…) in a population. I had long thought about how to do that but not until today did I begin to make some progress, so please allow me to share.

I’m going to make up a very-much simplified example using a Punnett square, something like this:


If you have no idea what this means, let this non-biologist try to explain as best I can:

Upper-case B and lower-case b in this diagram stand for two different versions of a particular (but mostly imaginary) gene that controls whether a person has blue eyes or brown. In this hypothetica example, the upper-case B gene causes brown eyes and is dominant, where the lower-case “b” causes eyes to be blue and is recessive. Thanks to the magic of sexual reproduction, you get two copies of each gene, 1 from Mom and one from Dad, whether they stick around and raise you or not. (You have two similar-but-not-identical copies of each chromosome except for the X and Y chromosomes; your two versions of each gene are carries in corresponding locations on each of the two chromosomes. If I got this right.)

If you have brown eyes, then your genes might be BB or they might be Bb or bB (same thing). If you have blue eyes, then you have bb genes for sure — again, in this hypothetical scenario.

This Punnett square shows the probability of what will happe if two parents who carry Bb genes have sex and produce offspring. It reminds me very much of how we use an area model to show that (X + Y)*(X + Y) equals X^2 + 2*X*Y + Y^2.

In any case, each of those parents carries Bb genes, and when the eggs and the sperm cells are manufactured inside the parent’s ovaries and testes, one or the other version of the gene is put inside, but not both. And it’s random. So since each parent has a Bb gene, its probability of passing along upper case B (brown) is 1/2 or 50%, as is the probability of passing along lower case b (blue eyes).

You can now find the probability of all of the outcomes shown in the interior of the diagram. The upper left hand corner is BB, pure brown eyes, with probability 1/4 because 1/2*1/2=1/4 and also in this case all of the sections really do have equal areas.

The upper right hand and lower left hand corners represent the Bb cross; the child will have brown eyes. The probability of a Bb cross is 1/4 plus 1/4, or 1/2.

The lower right hand corner is the region representing the probability of pure bb offspring  which have (recessive) blue eyes. The probability of bb is 1/4.

Now let us add a couple of features.

1. This is not just a single mom-dad pairing: this is a representation of an entire reproducing population where genes B and b are present, each 50% of the time.

2. Let us also pretend that the bb combination is fatal: not a single one of them survive to adulthood and to leave offspring. (This is a very extreme hypothetical example of how evolution operates. Normally Deleterious genes aren’t so uniformly fatal!) or alternatively, a breeder of plants or animals might decide to not permit any of the blue-eyed bb offspring to reproduce. Eugenicists used to advocate sterilizing anyone who exhibited harmful, recessive genes,in order to improve the remainder of the human race.

At first glance, You would think that this sort of genetic selection, either by artificial or natural means, would work very quickly, and that after just a few generations, the proportion of the population that was blue-eyed would vanish.

Jim Sandefur said no, it would take a really long time. I forgot the details, and just worked them out today. I’ll work out the details for you later when I have a larger screen. But:

Bottom line: even with this 100% culling of recessive genes, the proportion of blue eyes goes down as the harmonic series (1/X), where X is 4, then 5, 6, 7, 8, etc

So if the first generation has 1/4 (25%) blue eyes, and if every single individual with blue eyes is somehow prevented from reproducing, then the next generation will still carry the lower-case b gene 1/5 (20%) of the time.

And if children with blue eyes (bb) are still prevented from reproducing, the third generation will still pass on  the lower-case b gene one-sixth, or 16.67% of the time, and the next generation will pass on the lower-case b gene one-seventh (14.29%) of the time. The next generation passes on b genes one-eighth (12.50%) of the time, then one-ninth of the time (11.11%), then one-tenth of the time (10.00%) and so on. At first the decrease is pretty rapid, but after that it slows to a craw, and the world would never be entirely free of the pure bb. After 100 generations, there still would be 1/103 (almost 1%) of the population carrying genes that can pass on blue eyes.

At 25-30 years for a human population to reproduce, you are talking about 2,500 to 3,000 years!

However, the fraction of the population that actually is born with blue eyes apparent to everybody will fall much faster.  The proportions would be 1/4 in the initial generation, followed by 1/9, then 1/16, then 1/25, then 1/36, then 1/49, then 1/64, and so on, with the ratio being 1/X^2 (one over x-squared) rather than 1/x.

So, by 10 generations, under this hypothetical, 100%-effective sterilization or extermination regime, the proportion of the population with visible blue eyes would have fallen to 1/169, about six-tenths of a percent. However, the fraction of the population that still carries the genes for blue eyes would remain at 1/13 of the population, about 7.7% of the total.

However, perhaps conditions might flip-flop. In my hypothetical problem here, perhaps the conditions making blue eyes fatal would disappear after a number of generations. (Even if Hitler’s nasty 1000-year Reich would not have been enough to eradicate whatever enemy genes!) In fact, perhaps the reverse would be true: having brown eyes would be a fatal handicap under some conditions. Then the prevalence of blue eyes would rise to the fore in their place, but there would be an enormous die-off of all those who had brown eyes, which would mean the vast majority of the population. So all that would be left would be those formerly recessive genes, and the formerly dominant genes would be wiped out completely.

More realistically: recessive genes that make people susceptible to die from some particular disease or parasite or environmental factor do definitely get reduced in frequency over time, as I hope I have shown. However, they do not disappear completely for a very long, long time (if ever!) unless the entire population is reduced to just a handful of individuals, none of whom carry that gene, just by chance.

Evolution does work on those time scales. Human societies and any proposed eugenics program do not. Evolution has no direction, and is essentially blind, like a mathematical algorithm.

People often say that everything happens for a reason. Often, that reason is simply the laws of probability, which are extremely hard for most people to handle. Myself included.

 

2016 Hopewell Observatory Fall Open House and Star Party October 29/30

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2016 Hopewell Observatory

Spring Open House and Star Party

October 29/30

 
Anybody interested in the night sky, including members of local astro clubs like NCA and NOVAC, are invited to the Fall 2016 astronomical open house and star party at Hopewell Observatory on the night of October 29/30 (Saturday evening and on into Sunday morning), 2016. Feel free to pass this invitation to friends, neighbors, and family and anybody else you care to notify.

 

We are located about 30 miles west of the Beltway on Bull Run Mountain – a ridge that overlooks Haymarket VA from an elevation of 1100 feet, near the intersection of I-66 and US-15. Detailed directions are below.

 

Assuming good weather, you’ll get to see planets, star clusters and nebulae and the Milky Way itself, as well as many other galaxies. If you like, you can bring a picnic dinner and a blanket or folding chairs, and/or your own telescope, if you own one and feel like carrying it. We have outside 120VAC power, if you need it for your telescope drive, but you will need your own extension cord and plug strip. If you want to camp out or otherwise stay until dawn, feel free!

 

Warning: While we do have bottled drinking water (and will have hot water and the makings for tea, cocoa & coffee) and we do have hand sanitizer, we do not have running water; and, our “toilet” is of the composting variety. Plus, you may want to consider bug spray, since we are completely surrounded and protected by woods. Do check carefully for ticks when you get home. If you do apply insect repellent while visiting, please keep the spray downwind from anybody’s telescopes!!

 

The road up here is partly paved, and partly gravel or dirt. It’s suitable for any car except those with really low clearance, so leave your fancy sports car (if any) at home. Consider car-pooling, because we don’t have huge parking lots. We will have signs up at various places along the way to help guide you, and will try to have parking spaces denoted.

 

Two of our telescope mounts are permanently installed in the observatory under a roll-off roof. We have others that we roll out onto the grass in our roughly one-seventh-acre field. We have two 14-inch scopes (one hand-made Dob and one Celestron SCT), a 30-cm Wright-Newtonian entirely built by our oldest member, and a 10” f/9 reflecting scope also made by hand. The entire observatory was hand-built, and is maintained, by the labor of its founders and current members.

 

The drive is about an hour from DC. After parking at a cell-phone tower installation, you will need to hike about 100 yards to our observatory. Physically handicapped people, and any telescopes, can be dropped off at the observatory itself, and then the vehicle will need to go back to park near that tower. To look through some of the various telescopes you will need to climb some stairs or ladders, so keep that in mind when making your plans.

 

It’s not the inky-scary dark of the Chilean Atacama or the Rockies, but Hopewell Observatory is mostly surrounded by nature preserves maintained by the Bull Run Mountain Conservancy and other such agencies. Also, our Prince William and Fauquier neighbors and officials have done a pretty good job of insisting on smart lighting in the new developments around Haymarket and Gainesville, which benefits everybody. So, while there is a pretty bright eastern horizon because of DC and its VA suburbs, we can still see the Milky Way whenever it’s clear and moonless.

 

This will be the official night of New Moon for October 2016. Venus and Saturn will be setting soon after the Sun, which will set at 6:12 pm with real darkness (and the end of all twilight) holding off for about another hour. The sun will rise at about 7:36 the next morning. Mars will be low in the south-west at sunset, and Jupiter will rise shortly before dawn.

 

We should also be able to track down and examine many, many deep-sky objects, including the famous Andromeda galaxy and the Orion nebula.

You can find detailed directions and a map to the observatory below:

 

DIRECTIONS TO HOPEWELL OBSERVATORY:

[Note: if you have a GPS navigation app, then you can simply ask it to take you to 3804 Bull Run Mountain Road, The Plains, VA. That will get you to step 6, below.]

(1) From the Beltway, take I-66 west about 25 miles to US 15 (Exit 40) at Haymarket. At the light at the end of the ramp, turn left/south onto US 15. (Exit is at approximately latitude 38°49’00″N, longitude 77°38’15″W.)

(2) Go 0.25 mi; at the second light turn right/west onto VA Rt. 55. There is a Sheetz gas station & convenience store at this intersection, along with a CVS, a McDonald’s, and a Walmart-anchored shopping center on the NW corner. This is a good place to stop for restrooms or supplies.

(3) After 0.7 mi on Va 55, turn right (north) onto Antioch Rd., Rt. 681. You will pass entrances for Boy Scouts’ Camp Snyder and the Winery at La Grange. (38°49’12″N, 77°39’29″W)

(4) Follow Antioch Rd. to its end (3.2 mi), then turn left (west) onto Waterfall Rd. (Rt. 601), which will become Hopewell Rd. (38°51’32″N, 77°41’10″W)

(5) After 1.0 mi, bear right onto Bull Run Mountain Rd., Rt. 629 (this is beyond Mountain Rd.). This will be the third road on the right, after Mountain Rd. and Donna Marie Ct. (38°52’00″N, 77°42’08″W) Please note that Google Earth and Google Maps show a non-existent road, actually a power line, in between Donna Marie Ct. and Bull Run Mtn. Rd.

(6) In 0.9 mi, enter the driveway on the right, with the orange pipe gate. There is a locked stone and metal gate on the left, opposite our entrance, labeled 3804 Bull Run Mountain Road. Don’t take that road – it goes to an FAA radar dome. Instead, go to the right (east). We’ll have some signs up. This is a very sharp right hand turn. (38°52’36″N, 77°41’55″W)

(7) Follow the narrow paved road up the ridge to the cell phone tower station. You should park around the tower (any side is fine) or in the grassy area before the wooden sawhorse barrier. Then you should walk the remaining hundred meters to the observatory on foot. Be sure NOT to block the right-of-way for automobiles.

(8) If you are dropping off a scope or a handicapped person, move the wooden barrier out of the way temporarily, and drive along the grassy track to the right of the station, into the woods, continuing south, through (or around) a white metal bar gate. The few parking places among the trees near our operations cabin, the small house-like structure in the woods, are reserved for Observatory members. If you are dropping off a handicapped person or a telescope, please do so and then drive your car back and park near the cell phone tower.

Please watch out for pedestrians, especially children! The observatory itself is in the clearing a short distance ahead. We do not have streetlights, and there will not be any Moon to light your way, so a flashlight is a good idea. In the operations cabin we have a supply of red translucent plastic film and tape and rubber bands so that you can filter out everything but red wavelengths on your flashlight. This will help preserve everybody’s night vision. In the cabin we also have a visitor sign-in book; a supply of hot water; the makings of hot cocoa, tea, and instant coffee; hand sanitizer; as well as paper towels, plastic cups and spoons.

The location of the observatory is approximately latitude 38°52’12″N, longitude 77°41’54″W. The drive takes about 45 minutes from the Beltway. A map to the site follows. If you get lost, you can call me on my cell phone at 202 dash 262 dash 4274.

map to hopewell june 2016

Hitherto Un-noticed properties of Primes

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Did you know that if a prime number ends in a 9, then the next prime number larger than that is much more likely to end in a 1 than in a 9?

Did you know that if you study coin tosses, it will take you only about four tosses to find a head followed by a tail, but about six tosses to get two heads in a row, even though they are e qually likely?

I didn’t either. Interesting article:

https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/

A ‘Maker Faire’ in Reston, VA

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I went to a “mini maker Faire” in Reston mostly as an exhibitor (on telescope making, hence the home made scopes) but also had a bit of time to visit with other “makers” before the gates opened.

The silly little ears are from a gizmo that was supposed to read and interpret brain waves … either my head was too thick or I broke the gizmo or I am dead: It couldn’t read my brain waves.

It was fun, and I talked to a lot of very interesting people. I regret that I didn’t take Jeff Guerber up on his offer to staff my tables while I went and looked at the other exhibits. Two entire, enormous, modern NoVa public schools (a MS and a HS) were filled with exhibits on all kinds of crafts. Lots of 3D printers, physics stuff, including jugglers in my hall. I have no idea what was in the other ones!

I met a high school girl who had a T-shirt proclaiming her desire to become a mason. She told me she was well aware that there were very few (if any) female masons. I applaud her goal and hope she is successful!

Another recent scope at the NCA – ATM workshop at the CCCC

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This ultra-short scope, by Todd M, has a mirror of 4.25″ (108 mm) and a pretty short focal length – about 2 feet (60 cm). He made just about everything, right here in the NCA ATM workshop at the Chevy Chase Community Center. He ground, polished, figured, and even helped aluminize the primary mirror; made the primary cell AND the spider and secondary holder; made all of the rest of the mount that you see; and even made the focuser itself from some plumbing parts!

It’s a very nice job, meriting a lot of praise. In case you were wondering, the paint was a special, very-high quality and very expensive top-of-the-line alkyd enamel, costing about $200 per gallon – and we have two of them. Explanation: it was an ‘oops’ can that was specially ordered and mixed for someone who changed their mind and couldn’t return it. In exchange for a non-profit donation receipt in the name of NCA, Bill R was able to get the person to donate both gallons to us.

The spider and secondary holder are very similar to the one made by Ramona D that you can see here. The major differences are:

(1) Todd used busted bandsaw blades rather than steel strapping tape for the vanes. (Both were the same price: free.) After looking at both projects, which both turned out quite nicely, my conclusion is that if you want to use bandsaw blades, you have to heat-treat (anneal) them so they will have less of a tendency to break right at the location where you are trying to bend them by 45 degrees. (Heat it up to cherry red and then let it cool slowly in the air, making it softer and less brittle, I am told…)

(2) And of course, it certainly helps to grind down the teeth of the bandsaw blade both for safety and to reduce weird reflections. Strapping tape is about the same thickness as many band saw blades, but the tape is wider and hence more stable and less prone to turn crooked (I think).

(3) Todd used ordinary 1/4″-20 machine screws (aka bolts) to attach the vanes of the spider to and through the walls of the tube. He cut off the heads of the bolts and ground one side flat near the head, and then drilled a little hole in that flat part, tapped (threaded) that, and used a tiny little machine screw to attach the vane to the specially-prepared screw, in a process that I hope is clearly shown in these three drawings.

(4) Ramona, however, used thumbscrews instead of doing all that cutting, filing and tapping. Actually, our little tiny tapping drills didn’t play well with our bit holders – they kept slipping. So she just drilled holes in the center of each thumbscrew head, and bought three very small nuts and bolts and used them in the place of the little screw that Todd used.

(Thumbscrews like these:)

thumbscrews