Egg drop (but not the soup)

Eggs aren't just for use in radiators.  For our latest home STEM project we made devices to protect eggs when they are dropped.  T, M and I were each limited to 100 plastic "bendy" straws, two rolls of scotch tape, two sheets of paper towel, and a plastic cup.  We bartered these items around; for example I traded my paper towels and cup for more straws and tape. 

M cut up straws into short pieces and used them as filler, like styrofoam peanuts, inside the cup.  The egg was wrapped in a paper towel in the middle.  She taped that in then lined the sides of the cup with straws against impact.

T's design used the least straws.  He used two cups and had one cut smaller and inverted inside the other.  The smaller one could compress inside the larger and create a shock absorber effect.  He wrapped the egg in a paper towel inside and had a few straws to cut and place around the outside haphazardly.

In my design the top is a 20 sided icosahedron with the egg just slightly below the center to give a "down" weighting.  I taped across the top 10 faces to act as a parachute (with a small gap at the apex to help stabilize it).  I used the remaining tape and straws to add legs at various angles to the lowest point to break the initial impact and convert the falling motion into a roll away motion.   

It unintentionally turned out looking a little like a virus particle.

 Time to test them out.  We started off with short drops of 3 feet and 5 feet onto concrete. 

No eggs were broken so we had to move to higher drops.

Everything worked at seven feet.  My virus parachuted down and rolled away.  At nine feet T's contraption flew apart on impact, but the egg was unbroken!  However, M's egg finally broke at the nine feet drop. 

After this point we had to climb onto the garage roof to get greater height over the cement.   

Tossing it off the roof at 12 feet finally broke T's egg!  The virus egg did fine.  It parachuted down at the same speed as before.

T and M are already demanding a rematch.  They have new ideas now for modifications.

Egg in the radiator

I kept topping off the water in the radiator each day but the leak got steadily worse.  I asked around and had a mechanic highly recommended to me; however, I did not have enough money at the time and had to wait till payday which was several days away.  I started keeping extra water in the back of the van just in case and I made an appointment to fix the radiator on payday.  Then, on the way home, the leak suddenly became much worse and would hardly hold any water.  I had to stop several times on the way home and fill it up to keep the engine from overheating.  There was a steady stream of water coming out of the radiator.  I used up all my backup water in the van and barely made it home.  We only have one vehicle so there was not a second car I could use.  Payday was three days away so we were grounded for three days.  I called in at work and worked as much as possible from my computer at home.  However, there were two problems.  I had a doctors appointment the next day that I absolutely could not miss.  And, I had to somehow get the van to the mechanic which was many miles away on the other side of town.

I had heard about an old trick for leaky radiators--cracking an egg into the radiator--and looked it up online.  There is a lot of debate surrounding this (see here and here); some people say it might work and some say absolutely not.  So I tried it.  Actually I had T crack an egg into the radiator with a lot of water.  He thought I was nuts.  I wasn't so sure either.  I worried about clogging the water pump; clogging the heater core however was not so much of a concern since we never use the heater.  I started it up and let the engine run for a while.  The theory is as the water in the radiator heats up the egg cooks and solidifies and seals the leak as it is pulled through.  We tried driving down the road a short way and back and it seemed to work almost immediately.  The radiator was holding water and the engine was not overheating.  We popped the hood and there was a faint cooked egg smell.  The egg worked perfectly for the next few days.  I was able to drive to the doc appointment, and to the mechanic.  Finally on payday I had the radiator replaced and the system flushed out.

Waxbill

There are fast moving flocks of tiny waxbills (Estrilda astrild) here on Oahu.  M spotted some and I managed to get a picture (of her and the birds).

 
They are grass seed eaters native to east and southern Africa, but now they have spread to S. America, Europe and islands around the world.  Like the java sparrow they are in the Estrildid finch family, in fact I have seen Java sparrows joining and moving along in their flocks. 

Here is a java sparrow hanging out with the waxbills.

Often they weigh down the grass heads and feed on them next to the ground.

Also, they seem to spend a lot of time preening. 

sz

On to the heptagonal

I couldn't help keep thinking about the polygon "figurate" number patterns on the drive to work this morning.  After the hexagon comes the seven-sided heptagon.

But first lets review the pattern.

The triangular number series is generated by n(n-1)/2.

The square numbers are generated by n^2, but also as the sum of two consecutive triangular numbers n(n-1)/2 + n(n+1)/2, just as a square can geometrically be divided into two triangles. 

The pentagonal numbers are sums of two consecutive triangular numbers, except with two of the smaller number:  2(n(n-1)/2) + n(n+1)/2

The hexagonal numbers of sums of a square number of two triangular numbers, and the square component can be reduced to two triangulars.  This comes out to: 3(n(n-1)/2) + n(n+1)/2

The pattern emerging is, the figurate number series corresponding to an s sided polygon is (s-3)(n(n-1)/2) + n(n+1)/2

If this is true than the heptagonal numbers should be 4(n(n-1)/2) + n(n+1)/2.  This gives 1, 7, 18, 34, 55, ... and these are the heptagonal numbers!

Also, two of the triangles can be combined to a square number which gives:

 (s-4)(n(n-1)/2) + n^2

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At this point I did some more searching online; this seems so obvious it must be out there.  In fact there is a wikipedia page devoted to polygonial numbers that I missed before (I had been searching too specifically for "pentagonal numbers," "triangular numbers" or "figurate number").   Under formulae it says:

The n^th s-gonal number is also related to the triangular numbers T_n as follows:

which is equivalent to the general case I came up with above.

(s-3)(n(n-1)/2) + n(n+1)/2

Pentagonal and Triagonal Numbers

The last post got me to thinking.  If a square can be divided into two triangles, and a square number is the sum of two triangular numbers, is a pentagonal number the sum of one larger and two smaller triangular numbers?

The first pentagonal numbers are 1, 5, 12, 22, 35, ...  The triangular numbers are 1, 3, 6, 10, 15, ...  Sure enough, 5 = 3 + 1 +1, 12 = 6 + 3 + 3, 22 = 10 + 6 + 6, 35 = 15 + 10 + 10...  The series of pentagonal numbers are made up of consecutive triangular numbers, two of the smaller one and one of the larger. 

...then can a hexagonal number be reduced to a rectangular number and two equal triangular numbers?

The hexagonal numbers are 1, 6, 15, 28, 45, ...  And indeed 6 = 4 + 1 + 1, 15 = 9 + 3 +3, 28 = 16 + 6 + 6, 45 = 25 + 10 + 10, ...  The series of hexagonal numbers is equal to the sums two triangular numbers and a square number (a special case of a rectangular number).

From this point there are obvious ways to reduce the hexagon and further polygons.  Perhaps this connection to number series is obvious to a mathematician, and is well known in mathematics, but I have not heard of it. It is fun to come up with something new (at least new to me) that appears to work.

Triangles and Bridges

The kids and I have been working on some geometry.  One part of this is showing how polygons can be subdivided into triangles.  I had T work out the formula t = s - 2; the number of triangles, t, needed to build a polygon with s sides.

Also I showed the connection between reducing a square into two triangles to reducing a square number, 2x2=4, 3x3=9, 4x4=16, ... to two triangular numbers 4=3+1, 9=6+3, 16=10+6, where 1, 3, 6, 10, ... are triangular numbers (can be arranged in a triangle, see my earlier post about pairwise comparisons). 

I also tried to illustrate how a triangle is more stable than a square to bending.  This led to a bridge building experiment the next day.  I picked up two bags of gum drops and a box of toothpicks from the grocery store.  I gave the kids 100 toothpicks each (midway we increased this to 110) and told them to build a bridge between two blocks (borrowed from F) one foot apart.  Then we would test how much weight they could hold before buckling.  (As a kid I tried to do something like this in my grandma's kitchen with bread dough and toothpicks; I refined my initial idea by looking online and seeing the post about gumdrop bridges at Beck Logan's STEM Blog.) M immediately said she would use triangles because that's how the Eiffel Tower is built. 

The kids really got into it.  This was a lot of fun.

Getting Started.  M has a pyramid base up already. 

Some more shapes appear. 

Taking Shape

T went for a Warren Truss design, a weight efficient pattern used in airplanes.  He also has a stick and gumdrop bystander.

M went for a series of interconnected pyramids with a complex base on each block. 

Measuring the fit. 

T's completed bridge, using 103 toothpicks, supporting four pencil weights.

T's bridge supporting 10 pencils.  At 10 it is starting to bend under the weight, but is still holding. 

Structural failure at 11 pencil weights. 

M redesigned her bridge a bit with anchors and weights at the ends to try to keep it from sagging so much as it pulled toward the middle. 

Placing the fifth pencil. 

Failure at six pencils. 

I didn't want to be left out of the fun so I made a bridge that combined the two kids designs and used exactly 110 toothpicks.  A Warren Truss with a row of pyramids on top to support against compression of the top edge. 

The combined design holding 12 pencils.

Structural failure at 16 pencil weights!

Kealia Trail

We went for a hike last weekend on Kealia Trail.  The trail is behind an airport so it is imposing to get to.  First we had to drive in on an access road at the end of the runway, with signs that say no entry all over the place.  Then park next to the runway, with no fence separating us or anything from the planes.  Then the first part of the hike has "restricted area" signs all over the place.  Anyway, once you get past all of that it is a nice little hike. 

M found a large ant nest.  There are no native ants, bees or wasps in Hawai'i, but now they are everywhere.  We even had a nest of ants in our printer for a while, little smashed ants would be on some of the pages we printed out, but now they have moved elsewhere. 

The trail is very rocky.  Below M is "lifting up" a boulder to see what is under it.

We just went for a short hike, partway up.  I hiked the entire trail before the rest of the family moved here, it keeps climbing all the way to the end.  I didn't want to push the kids too much; I wanted them to enjoy it so it is easier to go for more hikes in the future.  Plus V was carrying F the whole way.