Hexagonal Blanket Dimensions


Look at the right-most figure in diagram. The small triangle defines the all the major dimensions of the hexagon. Assuming the user measures the edge-to-edge dimension $c$, he or she can calculate the rest of the measurements. A simple right-triangle expression gives the relationship between $s/2$ and $c/2$, and is readily solved for $s$ in terms of $c$,
$$ \frac{s}{2} = \frac{c}{2}\tan\left(30°\right) $$
$$  s = c\tan\left(30°\right) = c\frac{\sqrt{3}}{3}. $$

Similarly, the Pythagorean formula gives the relationship between $c$ and the two other dimensions,
$$  d = \sqrt{ c^2 + s^2} = \frac{2c}{\sqrt{3}}.$$

Simplifying this equation by substituting for $s$ into the previous equation and simplifying produces
$$  \text{long pitch} = s + \frac{d-s}{2} = \frac{s+d}{2}  = \frac{\sqrt{3}}{2}c \approx 0.87c. $$

A blanket $n$ hexagons by $m$ hexagons will be approximately
0.87c\,n \times m\,c, $$
where the $m$ and $n$ dimensions are as shown in the next figure.


If a blanket will be wider at the ends, as shown in the figure, then $n$ will be odd. The total number of hexagons will then be
$$ \text{number of hexagons} = m \frac{n+1}{2} + (m-1)\frac{n-1}{2}. $$

In the example there are $(n+1)/2=5$ tall columns and $(n-1)/2=4$ short columns. So there are 6×5=30 hexagons in tall columns and (6-1)×4=20 hexagons in the short columns, or 50 hexagons overall. Using equation for blanket size and assuming $c=10$ inches, the approximate dimensions of the finished blanket are 9×10×0.87 by 6×10, or 78.3 by 60 inches.


School, from our house as the crow flies, is 5.73 km. If we neglect air resistance and deal strictly with ballistic flight then we can materialize a wonderful fantasy. Starting in the backyard, extending over the top of the house, is a launch-o-rocket, a rail-like launcher that accelerates the school-bound student until he or she can cruise over the city and arrive without bother of traffic. Our charter is to find the acceleration of the student from the launch-o-rocket.


Finding the Initial Velocity

We rely on the well-known fact that the maximum distance in a throw occurs when the departure angle is 45°. The vertical speed and the horizontal speed are equal. We denote these two identical speeds as $s$. Since distance is time multiplied by speed, the distance from home to school $d$ is
$$  d = t\cdot s.$$

We know the distance $d = $ 5.73 km.

Turning to the vertical speed, the student departs the launch-o-rocket with vertical speed $s$, but is immediately subject to gravitational acceleration. Since the student’s upward flight is exactly matched by his or her downward flight. Because the flight is matched, the student spends $t/2$ time rising and $t/2$ time descending. Since the student has no vertical speed at the top, we know that his or her speed is
$$  s = g\frac{t}{2},$$

where $g$ is the gravitational acceleration 9.8 m/s2.

Now, we have a system of equations

$$  d = t\cdot s $$
$$  s = g\frac{t}{2}.$$
The system looks like it has a many variables, but really there are only two, $s$ and $t$. We know $g$ and $d$. To solve the system we substitute for $s$ in the first equation with the second to get

$$  d = tg\frac{t}{2} = g\frac{t^2}{2}$$
Solve for $t$
$$   t = \sqrt{\frac{2d}{g}}
= \sqrt{\frac{2\cdot 5.73\,\text{m}}{9.8\,\text{m/s}^2}}
\approx 34.2\,\text{s}.
Not a bad commute, a little over half a minute.

With $t$ in hand, we can find the magnitude of the initial velocity. Remember that the initial velocity is $s$ in the horizontal direction and $s$ in the vertical direction, so the speed when leaving the launcher is
\left| \mathbf{v}_0\right| = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}.
The initial speed the student must attain is given by the very first equation, $d = s\cdot t$. Solving for $s$ with the value of $t$ we found, we get

$$  s = \frac{5.73\,\text{km}}{34.2\,\text{s}} = 168\,\text{m/s}. $$

Finding the Acceleration

The ramp lives on a footprint that is about 80 ft, or 24.4 m. It is also 24.4 m tall, so special zoning is surely required! The rail of the launch-o-rocket is the hypotenuse of a triangle, and that triangle has sides 24.4 m, and a total length of $\sqrt{2}\cdot 24.4\,\text{m} = 34.5\,\text{m}$.

The formula for position after a period of acceleration is
$$   p = \frac{1}{2}a\tau^2.$$
For our system, we also know that the acceleration is the change in speed divided by the change in time. Our speed goes from zero to 168 m/s in $\tau$. Again, we have a system of equations,

$$  34.5\, \text{m} = \frac{1}{2}a\tau^2 $$
$$  a = \frac{168\,\text{m/s}}{\tau}.$$

Solve for $a$ by first solving the second equation for $\tau$, and then substituting that result into the first equation to get

$$   34.5\, \text{m} = \frac{1}{2}a\left(\frac{168\,\text{m/s}}{a}\right)^2 $$

$$   a = \frac{\left( 168\, \text{m/s}\right)^2}{2 \cdot 34.5\,\text{m}}
= 407\, \text{m/s}^2 = 41.5\, g. $$

The typical onset of death occurs when acceleration exceeds about $10g$, so unfortunately, the launch-o-rocket is a single try system.

300 Million Years Ago

This weekend my family came with me to hunt fossils in the Jemez mountains of New Mexico. We hunted fossils along Route 4, where I understand collecting is legal and ethical. The fossils we were looking for are quite common, mainly brachiopods (similar to clams), and crinoids (a kind of anemone). These fossils are common in the Pennsylvanian group, especially in the Madera subgroup. These strata were laid down in the Late Carboniferous period from about 323 to 299 million years ago.  

I had excellent fun assembling our fossil hunting map. I used the grand open source QGIS geographic information system software, importing the road layers from the Open Street Map project through QGIS’ built-in plug in. The topographic information came from the USGS 1 m digital elevation model, which I traced at 25 meter intervals. The New Mexico Geological Survey provided a very detailed geological map of the Jemez Springs area, from which I selected the Pennsylvanian Madera sections. The total map is assembled in the following graphic. 


My daughter found a crinoid calyx, or at least I believe that is what it is.20170730-145520-02

You can see the fan-like structure at the top of the crinoid tapering to one end where the frond-like structures joined the stem.


My son found a brachiopod, which was nearly completely isolated from the surrounding matrix, and it is nearly flawless without preparation.20170730-145621-04

I composited three perspectives of the same fossil to show all angles.


I found a collection of small sesame-seed sized bumps, which I believe are fusilinids. These are shells deposited by single-cell animals, which makes them gigantic as single-cell animals go. I’d like to cross-section one, since some fusilinids have really complex structures.


We also found an assortment of random bits piled together. It seems like a story of the past, I just can’t read. I don’t know what all the elements are, but I recognize sundry crinoid stem segments and shell pieces. I don’t know what the long stem-like structure is left of center, but it has a fascinating look.


We had fun, found far more fossils than I expected, of far more types. Next, I want to learn some amateur fossil preparation. So amazing to hold the remains of something that lived 300 million years ago in your hand.

Hot Soup!

My kids blow on soup to cool it down. Sometimes they blow like a tornado and most of the soup exits the spoon rather than entering the child. I always tell them to blow gently—it will cool just as fast.

Well, we put the theory to the test. For cleanliness, we simulated a broth soup with hot water. Viscous soups, like cream soups, might be different.


For apparatus we mounted a spoon in a third hand tool. A thermistor temperature sensor was affixed to have the sensor at the lowest point in the spoon’s bowl. We had a small hand-held anemometer to measure the air speed at the spoon. We transferred hot—nearly boiling—water into the spoon with the turkey baster. The data logger, my own design, recorded the temperature.


Each of three test subjects blew on the soup to cool it. Each person blew lightly or strongly. We attempted to calibrate the airspeed at the spoon during an exhale. The picture below shows the first test subject (Dear Daughter) measuring the air speed. The anemometer is aligned with the spoon. Each subject attempts to hold their head in the same position.



Each subject blew on the spoon until the thermistor reading fell below 30°C. We recorded the start time and the stop time as indicated by the data logger. The start and stop time became cut points in the data processing.


We did all the experiments over about an hour, and left the data logger running the whole time. The plot of temperature is shown next. The spikes correspond to times when we recharged the spoon with hot water. The drops at the bottom of each trough correspond to our evacuation of the remaining water from the spoon.


A cut portion of the timeline, corresponding to each experiment, is fitted to an exponential function of the form Temp = A×exp(time×B). The time constant for these fits is –1/B. The time constant corresponds to the amount of time for the temperature to drop 73% of the way to the ambient temperature.


The time constants from each trial also describe a curve as a function of the air speed. I fitted these data with another exponential function, and although the fit is not exact, it is satisfying close to the data.


Each point color in the graph represents one person’s attempt. The dark blue point in the upper left is the a control where nobody blew on the spoon. The red point in the lower right corner corresponds to my son’s fastest effort, which removed about half the soup from the spoon.

To summarize the results, faster air speed does cool the soup considerably faster—meaning that I was not correct. My recommendation, based on the data, is to blow as fast as possible provided the soup is not sloshing out. You are welcome to use this data next time you’re trying to convince a young child to cool their soup properly. Your mileage may vary.

Rocket Stove

Maybe you’ve stored 100 pounds of wheat kernels, 75 pounds of beans, barrels of water and you’re all set for whatever municipal unreliability may bring. One question:

How will you cook the food you’ve stored?

In Food Storage and Refried Beans, back in April 2009,  I determined that to cook 100 pounds of beans would take about 4.7 gallons of white gas (Coleman fuel). Assuming the rest of a typical day takes a little less gas, you would still have to store about 10 gallons of gas to service that much food. While this is doable, it is certainly a hazardous amount of fuel. My personal solution is my own take on the Rocket Stove.


We have a wood burning fireplace, but it is not set up for cooking. However, we keep wood around for the cold winter nights. Obviously, mankind has cooked on wood basically forever. So, obviously it is possible. I was introduced to the rocket stove developed by Aprovecho, which is intended to reduce fuel consumption by more efficiently transferring the wood’s energy into the cookpot, reduce air smoke and soot inhalation by combusting more efficiently, and improve burn safety. They have ten design principals, summarized:

  1. Insulate around the firebox. I do this by using insulating kiln brick (about 0.65 g/cm3).
  2. Place an insulating short chimney directly above the fire. Mine is about 9 inches high, made by two races of insulating brick.
  3. Heat and burn (just) the tips of the sticks. The shortness of the firebox accomplishes.
  4. Heat is regulated by the amount of fuel. I have no damper, which their research shows is not effective.
  5. Maintain a good fast draft through the fire. This is accomplished both with the chimney and a recovered steel grate that creates an air channel under the burning fuel.
  6. Too little draft makes excess smoke. See principle 7.
  7. The opening, size of spaces, and chimney should all be about the same size. Specifically they recommend a 12 cm square opening (4.75 in), which is about 3.5 inches, and is probably slightly too small.
  8. Use a grate under the fire. See principle 5.
  9. Insulate the heat flow path. My entire structure is made of insulating kiln brick.
  10. Maximize heat transfer to the pot with properly sized gaps. I have not yet begun this phase of development.

For an initial design I used twelve insulating kiln bricks. Four make a floor—insulated enough, I think, to be used on a wooden stand. Two were cut into plugs to make the sides complete, and the remainder were stacked to make a square chimney and burn chamber. The burn chamber as seen through the chimney is filled with embers.


I scrounged and bent a wire rack to make a grate that retains a channel for air flow under the combustibles.


On top I used four small stones to make a burner. This is very much the wrong design for quickly heating water, but it worked for a test run.


Actually, it worked for three test runs. Yesterday we boiled water for tea.


This morning the kids helped make oatmeal on it. This evening I caramelized onions on it while grilling burgers.

My impression is that this device is the bee’s knees! The smoke was minimal (not as minimal as I would have liked, but largely avoidable). The amount of wood burned was about 4 linear feet of thumb-diameter sticks. The stove is stable, even with my crummy burner. I think it would make a nice patio fireplace for autumn evenings. Small, but controlling the smoke makes it much more pleasant to be around.

I do look forward to improving it. It is a pain to move, since it is about 45 pounds and doesn’t hold itself together. It is too low to cook on comfortably. Its heat transfer are near the pot is not well sized. This all requires work. I would also like to measure the efficiency of the stove. Always more fun to have!

Masterbuilt Smoker Upgrade–Ribs Hanger

I love ribs, and I love the way they come out of my Masterbuilt electric smoker. Actually, it’s my dad’s smoker but I keep it safely at my house. The problem with ribs is the time. The late-night clean up, after cleaning and rubbing the ribs the night before and cooking six hours, is almost too much. The worst thing to clean is the racks. The chromed welded-mesh racks (picture below) sit in the smoker like oven racks. The racks’ many textures make them difficult to clean. My best success is to spray them with cooking spray before using, then soak them in the sink for an hour or two, followed by a brushing, and then often steel wool. They don’t get clean enough in the dishwasher.


This post is about making ribs without the cleanup headache.

My friend Scott owns the same kind of smoker, and provided the first piece of wise council. Instead of lining the water pan and drip pan with foil, put a disposable pan aluminum pan in place of the water pan. I also hang aluminum foil skirts on the sides of the smoker to keep from having to clean the sides as shown below. I use a 10.5×13 inch disposable pan purchased at Costco. I deform the pans a little to fit better.


To dispense with cleaning the racks demands an alternative. I built a hanger to suspend the ribs from near the ceiling. Then, I bent cheap stainless skewers into a shape that holds the ribs along their length. One skewer is native, unbent. The other has the end bent up and then into a loop. Then the two skewers interlock. String tied at the top holds the ribs in.



Each pair of skewers holds a half rack of ribs, squeezed. My design keeps the ribs upright with minimal risk of falling.


I align the loops at the top of the skewers so that they are both held.



I built a hanger out of piece of brass angle iron. I drilled and tapped along the length, and ran 2-inch 8-32 screws through, leaving plenty of length to hang the holders from. The rack fits snuggly inside the smoker, in fact too snuggly. However, the idea of the design seems sound. The angle iron rests on the wire slides Masterbuilt included. I use a steel clip. If I redo this, I’ll probably use a threaded hole on the inside of the wire slide, and a fender washer to hold the angle iron on.


To load the smoker, I hang the ribs from the screws.


To sauce the ribs, I use tongs to pull a whole rack out. The large rings in the skewer make them easy to take off for a saucing.



The skewers’ simple geometry makes them a breeze to clean. They came off the ribs without damaging the crust we work so hard to get. In addition to the easy clean-up, the method causes the fat to drip off the ribs and into a pan, rather than onto the rack of ribs below. The angle-iron hanger rack also works to hang chicken halves with a steel s-hook or a string, also with easy clean-up.

Useful Range of Vivitar Wireless Remote Release

Vivitar brands a radio-frequency wireless remote that is available on a budget. The version shown below is configured for a Nikon D300; the pigtail can be changed for other camera designs. The remote has a telescoping antenna that extends to about six inches. The radio receiver is about 1.5 by 2 inches as seen from the top, and stands off the table about 3/4 inch. The receiver has a plastic foot that fits in the camera’s hot shoe.


For an upcoming shoot of a dance performance my two-person team is planning to work one camera dynamically up close, working on solo dancer shots and facial expressions. A second photographer will be in the mezzanine working context and group figures, as well as providing a higher angle. In our last shoot I worked close to the stage, mainly on single-dancer shots. Usually a single-person subject looks best when her background is simple and clear. During group figures I could find no position near the stage that produced satisfying composition. In other words, during large group shots I had the choice of scrambling to a different perspective or sitting idle. I want a different choice.

The new choice is to put a third camera either on the mezzanine left, or better on the balcony. With a wireless remote I can easily capture a full context shot. I can set the balcony cam with a fixed focal length lens, fixed aperture, and fixed focus point to get a consistent context shot. When the stage situation demands context shots, I’ll drop back and work the remote. Assuming the remote works.

Two main issues could get in the way of the remote. The first is batteries. Both the remote and the receiver use a battery. Should be easily solved with new batteries prior to the shoot. The second one is distance; the remote must have enough range. I used a simple test to gauge the remote distance. My son stood by the camera on the sidewalk, and I walked a few steps at a time up the sidewalk. He would give me a thumbs up if the camera shutter released, and I could take another few steps further. I marked the final distance with chalk. We performed the test with the remote’s antenna collapsed, and with it extended. An outdoor test means the RF has little chance to scatter off walls and ceilings. I expect the remote’s indoor range will exceed its outdoor range.

Antenna Collapsed: 53 feet max range
Antenna Extended: 148 feet max range


Thistle and The Bear on Autumn and Halloween

Thistle and the Bear talk about autumn, and especially about Halloween traditions.



My seven year old daughter practices her typing with a writing prompt most days. When asked to type about what she likes about Halloween, the wrote this:

I like Halloween because we have a funny dinner and we go trick-or-treating in the dark. That’s fun to do because we are out of the house at night and ringing people’s doorbells and collecting candy in our bags that hold candy lasting til Christmas. We are usually out trick-or-treating for a long time in the night. I like being a mouse.

Container Gardening

Container gardening is ideal for the small suburban backyard without much space. It is even better suited to the arid climate because the required water is much less—in my system I water less than twice a week and use less than a gallon. I’m using a container gardening system worked out by others, and it is really impressive. My father put me onto good video documentation as well as EarthTainer’s excellent guide for soil composition. The video discusses using aluminized bubble wrap, sold as insulation. I did not initially wrap my buckets in insulation. An experiment is in order.


My tomatoes are dying. The leaves have started drying, browning, and curling. Older leaves are failing first. My diagnosis is an untreatable fungal infection called Verticillium wilt fungus or Fusarium wilt fungus. The symptoms, according to gardeningknowhow.com, include worsening wilting after watering. My cherry tomato has produced three fruits, none of which are likely to be eatable. Look carefully, you can see a hornworm with more hope than promise.


Apparently soil temperature can influence the spread of Fusarium. Temperatures above 30 C (86 F) contribute to the spread.

I put a thermal probe in each of my two containers for a day. After a day I wrapped one of them in insulation and placed white foam core on top. The buckets are in different positions, so I’m not looking at the difference from one bucket to the next, but rather the change in the difference. I plotted the soil temperature in each bucket over the course of the experiment, along with the air temperature.


The result shows that the insulation helps quite a lot. It lowers the peak container temperature by 3 to 4 degrees F. More importantly, it kept the soil temperature below 80F despite higher air temperature; that should help control fungal growth. In the New Mexico High Desert the insulation should not be considered optional. Partial shade may also be beneficial.