Carnival Of Math February 2020

A Collection Of Math and Science Blog Posts From Around The World

Final Look at 2019: School, Science and Education
by Frederick Koh
A detailed review of 2019 examining science, school and education related events.

Hypot – A story of a ‘simple’ function
by Mike Croucher
Even the most simple looking mathematical functions can be difficult to implement on computers perfectly.  In this post, I look at an extremely common computation where the mathematics can be understood by children and yet efficient and bug-free implementation is complex and the subject of modern research.

Convergence rate of random walks
by John Cook
In some cases, random walks rapidly become more uniformly distributed, quickly going from obviously not uniform to apparently uniform.

Attracted to Attractors
by Ari Rubinsztejn
In this post 3 different chaotic attractions are visualized.

More Modular Knitting
by Pat Ashforth
Geometry in knitting (even for those who ‘can’t do maths’). How many different shapes can be knitted using only 45, 90 and 135 degree angles?

The Multiples of Me
by Sam Hartburn
The Multiples of Me is a poem about prime numbers, and why they needn’t be sad about having no factors.

Two dimensional tessellations at the Curious Minds Club
by Debbie Pledge
I run a recreational maths after school club in England. The post shows I got the children to explore the regular and semi-regular tessellations.

Australian Mathematicians
by LThMath
At the start of January we wanted to do something on our Facebook page to raise awareness about all the problems Australia has been through in the last period. We were shocked at the situation there.  For 2 weeks we researched and wrote more about Australian mathematicians and their work. In addition, each post has a link where you can donate for different charities and organizations. In this post we want to put together all the information we have discovered in those 2 weeks, including the mathematicians and where you can still donate to help.

Welcome to a Carnival of Mathematics!

This month, I will be hosting the Carnival Of Mathematics blog.

The Carnival of Mathematics is a monthly blogging round up hosted by a different blog each month. The Aperiodical will be taking responsibility for organizing a host each month, and links to the monthly posts will be added here. To volunteer to host a forthcoming Carnival (see below for months needing a host), please contact them on their website.

The Carnival of Mathematics accepts any mathematics-related blog posts, YouTube videos or other online content posted during the month: explanations of serious mathematics, puzzles, writing about mathematics education, mathematical anecdotes, refutations of bad mathematics, applications, reviews, etc. Sufficiently mathematized portions of other disciplines are also acceptable.

A FAQ can be found HERE.

I have the honor of hosting the anniversary Carnival! The Carnival of Mathematics will be 13 years old on February 9th.

If you want to get your math related post submitted, fill out this Google Form for consideration.

Brace yourself, there will be a test later.

A Discovery of Bristlecones

Yesterday, we took a trip to visit the Ancient Bristlecone Pine Forest in the high Sierra mountains, in California. The forest is home to thousands of Great Basin Bristlecone Pines (Pinus longaeva) the oldest living trees on Earth. The oldest, clocking in at 4,851 years old, is named Methuselah. 

The altitude of the forest is high, about 10,120 feet at the beginning of the trail through the grove. It is illegal for pilots to fly much higher without supplemental oxygen. As soon as I got out of the car, I was winded. I didn’t stop feeling winded until I was there for about an hour.

The trail is 4.8 miles long and it took more than 5 hours to walk it. There were a few primary obstacles. One is the fact that I am 57 and I don’t trail run anymore. The other is we were constantly stopped by awesome vistas and spectacular photographic opportunities. I took 353 photos – that is 73 photos for every mile walked. Seventy-three!

At every turn there was a spectacular tree – completely different from the rest. Each tree has a different personality and different shape. And there are thousands of them. And most are thousands of years old. There is one valley where all the bristlecone pines were older than Plato’s Republic.

All of these are older than Plato’s Republic.

A bristlecone pine is shaped and sculpted by environmental and geologic forces. Fire will strip away its bark and then 500-800 years later it will rebud new growth. Wind and fierce winter furies will bend and twist the the trees into alien shapes. The rocky, inhospitable alkaline soil will twist and gnarl the roots into complex spirals. The tree rings are so fine they are about as thick as a human hair and must be counted via a microscope. If you put your ear up to the tree and knock on the wood, the sound is an eerie mixture of echo and underwater acoustics. The feel of the wood is near rock solid – with no discernible give or flexibility. It might as well be rock.

Each tree is a different and amazing, nature-made Bonsai tree on a very large scale. My 73 photos per mile were not enough to document them all.

Lastly, I did find Methuselah, the oldest documented living tree on Earth – and She was glorious. She is the true Goddess of the Grove. For an old Druid like me, it is a wonder to behold. We took a bunch of photos and selfies, tree-hugged a bit, whispered to Her and accepted Her blessings of peace. I will never share these photos or share Her position. Her legacy must continue after me as she has stood the ages as well as Her siblings surrounding her. Because you know, there is always that one asshole.

The above photo is one that we dubbed ‘Butthuselah” for obvious reasons.

Bag End 2

Econ-friendly Home Design Inspired By J.R.R. Tolkien

I have always been interested in bio-friendly housing, specifically designs that approach zero-environmental impact.

After seeing The Fellowship of the Ring, you have probably fantasized about living in a Hobbit Hole and lazing about in the shade. I know I have. The night I walked out of the theater on opening night, my OCD kicked in and I designed a Hobbit Hole.

That is when I started expressing my inner architect and wondering of easy, bio-friendly ways to build a Hobbit Hole. This is what I have come up with.

The largest expenses in building a home (not counting the flat screen tv and indoor lap pool) typically are the walls, exterior and roofing system. Obviously, the roof and exterior are done away with for Hobbit Holes. However you have some staggering stress and loading issues with underground housing. The weight of the soil and flora growing on it can produce tremendous loads on a structure. It is even worse when it rains.

Once you start doing the math for wooden structures, the cost quickly skyrockets. On top of high costs to support such loading, you have yet to deal with the issue of water seepage, insect vulnerabilities (termites) and wood rot.

That leaves us the two building materials. Steel and concrete. With the circular nature of Hobbit Holes, one could use large steel pipe, but no source exists that makes low cost steel structures that I could find. But concrete… yes… concrete is the ticket.

All around the world, companies manufacture pre-formed concrete pipe… LARGE concrete pipe. Concrete pipe has several advantages for making Hobbit Holes. They come with an assortment of flanges, protrusions and options that allows for the creation of windows, skylights, doors, garages, fireplaces and chimneys.

As one can see, using such materials for the construction of a Hobbit Hole would make for a comfy home. Or at least a conversation piece.

Although the elliptical may have some aesthetic advantages, the use of round concrete pipe may be the best choice. The main reason is that you will need to have room to place plumbing, ventilation, electrical and communication hardware though out the Hole.

The best solution for this is the area under your flooring. The best example of this type of construction is in naval architecture with sailboats. Plan all of your plumbing, ventilation and wiring, then figure out how much space you will need, vertically. Maybe the use of multiple styles, circular for halls and elliptical for rooms, is the best idea. The halls would need the most space for plumbing, etc as all rooms connect to it. That is the design philosophy I have used for Bag End 2.

The proper name for this type of home is “earth-sheltered home.”

There are some other similar designs and building strategies for constructing homes of this type.

Rammed Earth” homes are homes that are built using spare tires filled with packed dirt. The upside is that are recycling old tires, glass bottles and other uncommon materials to build a house and low cost for heating and cooling. The down side is the incredible labor involved with packing thousands of tires with dirt. Also, there is some concern with rammed earth homes in areas frequented by earthquakes.

Earth Ship” is another type of house. These may include rammed earth construction, but the main goal is off-grid living. Typically, these homes are found the the American West and are of an adobe design.

A Guide To Getting An A&P

My work experience prior to Baker’s

My father had a unique way to get me to learn about cars. He wouldn’t let me get my driver’s license until I learned how to work on cars. And what better way to achieve that than buy a Triumph TR7 that didn’t run. I remember the day it was TOWED to the house. There was to be no driving in my future until that car was fully repaired. Six months later, after a teardown of the engine to replace the head gasket, replacement of the transmission, the rear end, and the drive shaft. Keeping the Zenith Stromberg updraft crabs synced was an ongoing nightmare and DO NOT get me started on Lucas electrical system, which must have been conceived during a Welsh Demon Summoning ritual that went awry.

At the time all of that turmoil was going on, I was working for the local FBO, getting all of my civil time in for when I go get my A&P. At the same time I earned a 2 year degree in electronics and that helped greatly when it came to take the writtens, oral and practical.

But…. life happened. Those halcyon days of working on airplanes ended 25 years ago. I had my time in but I had forgotten a lot of what I had learned via on-the-job training.

When 2018 rolled around, I wanted to go back and finally get my A&P. I saw two ads for A&P prep schools in Trade-A-Plane. I always grab a copy of TAP whenever I can and one ad was always there; Baker’s School Of Aeronautics. I cut it out and saved it. I remember seeing this ad for YEARS in TAP. When I started asking around, Baker’s kept getting highly recommended, especially by a very good friend of mine that got his A&P from there. For me, his recommendation and the ARMY of other graduates that swear by Baker’s, my decision was finalized – Baker’s it is!

My Baker’s Prep

Since it was such a long time since I have wrenched on a certified aircraft, I went whole hog on training materials. I used the King Schools video course which was excellent for me because it actually shows what the questions are pertaining to visually. I get a lot out of visual presentation.

I also used the test prep apps from Dauntless Aviation. The flash card feature is excellent. The apps have excellent descriptions for those of use that need that visual of “how it works.”

For the King Schools video, I didn’t bother with the sample tests, I only watched the videos. Same with the apps, I only used the flash card feature. As the instructors at Baker’s will tell you, ONLY consume the correct answers. Recognize the correct answers and ignore the wrong ones.

List Of Things Not To Do Before You Arrive At Baker’s

  • Don’t take sample tests or random tests.
  • Don’t use the time before Baker’s to slack off – you study.
  • Don’t think this will be easy, it won’t be.

List Of Things Not To Do After You Arrive At Baker’s

  • Don’t go to Nashville – you study.
  • Don’t take weekends off – you study.
  • Don’t fart around after classes – you study.

The staff is there to help you succeed. They are a great bunch of people who are actually dedicated to getting you an A&P certification (don’t call it a license ffs). Don’t fear the DME’s they are very helpful and are not there to make you fail. 

Study Groups

Once you are into the oral and practical portion of the course, have your ass in a study group every night. EVERY. SINGLE. NIGHT. The act of asking each other questions is VITAL to succeeding with your orals. Nothing, no other study method can replace it. If you want to be the loner and study on your own, make sure you have the time off of work and the cash to stay a few extra weeks in Lebanon.


No matter what you spend getting your certs, it will still be waaaay cheaper than a semester of college. That said, there are loads of ways to lower your costs before you get to Lebanon and while you are there.

Some hotels are more expensive than others. I stayed in both the Hampton Inn and the Holiday Inn Express and they both have their pros and cons. I can only speak about the other ones via word-of-mouth from my fellow students.

Hampton Inn


  • Nice accommodations
  • Big lobby to host study groups
  • Good breakfast
  • Free snack pack during the week (bottle of tea, granola bar, coffee cake and a piece of fruit.
  • Literally next door to the school so you might be able to do away with having a car.


  • It is the most expensive choice of the school’s recommended hotels.

Holiday Inn Express


  • Nice accommodations
  • Big lobby to host study groups
  • GREAT breakfast options – it has a CINNABON bar! 
  • It is very affordable


  • You will probably need to rent a car or budget for a lot of Uber rides.

Love Your Walmart

It will cost you a lot if you eat out for every meal. Go to Walmart and buy food that is easy to prepare with just a small refrigerator and microwave. One of my fellow students ate baloney sandwiches for his entire time there.

Aviation Museums

If you are fortunate to live near an aviation museum, do yourself a favor and get a season pass. I went damn near every week for three months to learn all I could about turbine engines. I took hundreds of reference photos and studied them for hours when I would get home. I never worked on one and had a lot to learn to get through the powerplant written test as well as the oral and practical.

My Tips

If you don’t have any knowledge about electricity or electronics, I suggest getting this, an electronic education kit (Amazon Link). It has 130 different experiments that will teach you everything you will need to know about the fundamentals that you will face on the writtens, the orals and the practicals. For only $35, it is a great deal.

If you are a turbine guy and are unfamiliar with the archaic technology called a carburetor, consider this Army educational film on YouTube. Here are some other helpful videos…

Above and beyond all of the above, there was one thing that made me successful and that was flash cards. Not the flash cards from a software app, but actual notecards.

When I got the Oral and Practical book from Baker’s, I took each question, copied it by hand on the front of a notecard and put the answer in my own words on the back. I did this for most of the book for the questions I didn’t know like the back of my hand. When I wasn’t in group, I was going through this stack of cards, read each card and tried to answer. Once I did, I flipped it over to see if I got it or not. I would then place the card in a pile and move on to the card.

I did this over and over and over. Once I knew the card by heart, I would place it in a different pile and then continue on with the pile that I still didn’t know. I repeated this until I was left with no cards in my hands. I then repeated this up until the day I took my orals.


Of my class, only three zipped through everything on the first try. Most failed something; a written, part of the oral or the practical – me included. The powerplant written was a particular struggle due to my unfamiliarity of turbine engines.

Doing the prep work that I have suggested BEFORE you get to Baker’s will help you find success. If you don’t do the prep work and don’t put in the insane hours required to succeed, you can’t blame Baker’s. You have to do your job before they can do their’s.


Baker’s has a damn fine program to get you your certificate – best in the nation. Listen to your instructors. Work your ass off and you will get there.

My Closest Encounter

It was a September day in 1992 when me and some close friends were camping in Rock Castle Gorge near Floyd, Virginia.

Whenever I go camping I tote along a decent sized pair of binoculars for stargazing. I am not looking at anything particular, just looking. When it comes to stargazing, I am more of a casual tourist.

This Saturday night in September I saw something I had never seen before, something that should not and could not be there.

I first noticed a blinking light, to the north, about 330 degrees. It was high up. Very high up, well above normal commercial aircraft flight levels maybe 100,000 feet AGL. The blinking was maybe less the a one second cycle. The light was stationary relative to my position for about an hour, continuously blinking. At the end of the viewing, the blinking stopped, the object increased luminosity and then moved away astonishingly quick, up from its current position until it faded completely from view. If it had any tint to the color of the light it may have been blue, but I am unsure.

At first I thought maybe a quasar. “Do quasars give off radiation in the visible light band?” I thought to myself.  But the star field moved BEHIND it. The blinking light was holding steady. 

There was no evidence of it being an aircraft. There were no navigation lights and there wasn’t any evidence of a red rotating beacon. If it was a strobe, why were there no other lights? And why was it stationary? And I doubt there was some chucklehead pilot aboard flipping the landing light on and off for over an hour with perfect timing.

It wasn’t a helicopter. They can’t fly that high up. There isn’t enough air for rotor blades to maintain lift that high up. Besides, the object was about 4 times higher than the highest altitude helicopters currently available.

It wasn’t a meteor as it stayed stationary for over an hour.

It wasn’t a satellite. They move unless it is a geosynchronous satellite, those might blink to the observer and would remain in the same position in the sky, but they would only be visible to the south, behind me. I was looking the other direction.

So what was it? Here is what is wasn’t:

  • Quasar
  • Airplane
  • Helicopter
  • Meteor
  • Satellite

This clearly puts it as a UFO and in my opinion an extraterrestrial craft.

I am a big fan of Carl Sagan and he shaped my scientific view tremendously. He was often found to be advocating, “tremendous claims require tremendous evidence.” I totally agree. I am just one guy with one observation. But still, if you apply Occam’s Razor that stipulates all things being equal, the simplest explanation is usually the right one, you end up with an extraterrestrial craft.

If you apply Occam’s Razor to this observation, here are the competing ideas.

Quasar – if a quasar was rotating near Earth, within the orbit of the Moon, none of us would be alive right now.

Aircraft – if it was an airplane, the hovering and subsequent acceleration would have killed all aboard and resulted in a fiery crash.

Meteor – if it was a meteor that could stop and go, it would mean the meteor was under controlled navigation and meteors are just hunks of rock… that don’t blink.

Satellite – these objects move in a predetermined path, they do not hover, nor accelerate in other directions.

That leaves us with something from another world. They could have been scientists, observing mankind, or listening to Blue Oyster Cult’s (Don’t Fear) The Reaper on WROV 96.3 FM, Roanoke’s source for classic rock – The Rock Of Virginia!

But maybe they were just out and about. Maybe they were a lot like me, casual tourists in a very large universe.

The Day I Met Carl Sagan

Carl Sagan

I was born in 1963 and grew up on large doses of Apollo. Anything and everything that revolved around NASA, I consumed it with a hunger only a geek could produce. During the 70s my family lived in West Virginia and we would take these trips to various places in the state; Blackwater Falls, Dolly Sods, Pipestem – just about anyplace where we could hike, do a bit of fishing and pitch a tent.

On one such trip we took a wrong turn and happened upon the Green Bank Radio Observatory. My head exploded. I had no clue such a facility was in my home state.

There was a tour offered so we stopped in and took a short bus tour of the facility – the guide pointing out the frolicking deer, the softball field and the radio dishes as we went.

The last stop on the tour was in a little meeting room where we watched a short film and heard a presentation. The presenter, with a very distinctive voice, introduced a brand new film titled “The Power Of 10.” After the film, he gave us a short introduction to a program called SETI. If my young kid brain wasn’t blown by then it was blown now. I had never even considered that other civilizations could be using radio. At that period in my life I was an avid ham radio nerd and knew it was possible to pick up on interstellar signals. Only in ham radio, we called such signals “noise.” Whoever that guy was on the last stop of the tour gave me a lot to ponder.

A couple years later a new TV show about space titled “Cosmos” was announced and I was thrilled! Like the geek that I was, I actually had a notebook so I could take notes as the show went along. As the narration started I instantly knew the voice. Lo’ and behold, here was that same guy from Green Bank now on television telling us how Earth was the shore of the cosmic ocean.

So, I met Carl Sagan before he was “Carl Sagan.” I remember at the presentation he was enthusiastic and actually engaged with the tourists. Even then, his passion for teaching was memorable.

I really lucked out. I know this was not his day job. I am sure he was just at the observatory and couldn’t pass up a chance to give a new group of people a new insight to the stars.