Sunrise and Baloney

 
I am sorry to tell you that today was pretty much a non-starter. We did perform bell choir this morning, doing Pachelbel's Canon in D. It's the hardest piece we've done this year, but also a very pretty arrangement. We did not embarrass ourselves.

The sun was shining, and I should have at least gone out for a hike if I couldn't make myself work in the afternoon. But the truth is that I didn't sleep last night, and I just couldn't make myself get moving. I played my game and read a book. Oh, and I washed the dishes. Big deal.

Enjoy two very different sunrises from the week. This is Wednesday.



This is Saturday. I'm not going to apologize for something as "ordinary" as a sunrise this time. I read this week that people don't look at the sky very much. I find that hard to believe. At least the people I am friends with on Facebook post lots of beautiful pictures of the sky. Have a couple more. For sure it is two more days I got to have a sunrise, and I appreciate those more and more all the time.



In the genuine baloney category, you know how the memes were saying that the palindromic date of 02/02/2020 would not be repeated for umpity-ump years. People just don't think for themselves. I didn't see anyone post that it will happen again next year on 12/02/2021. The memes also said it hadn't happened for 900 years or some such nonsense.

Remember 11/02/2011? That was a nice day too. I was contemplating beautiful fungus.



Or 01/02/2010? Ellen and I went to Bowman Lake and snowshoed.



Or 10/02/2001? That was before I was blogging, but I know what I was doing. Marie and I were hiking on the Superior Hiking Trail.



And soon I'm going to contemplate the inside of my eyelids. I should be able to sleep tonight.

See Earthstar

Intersecting Curves

 
One morning earlier this week it was frosty outside and I thought this was an intriguing pattern. Notice how all three lines intersect at the same point in the lower corners.



Have you figured out what this is?

If you have, your mind correctly interpreted that although the lines appear to touch and be in the same plane, it really has three dimensions.

It's my ordinary old-fashioned clothesline. The lines don't really intersect because we know there is a third dimension. If we move to a different location in that third dimension, depth, the lines are actually pretty much parallel (and not nearly so interesting).



Gives a person something to think about, eh? We live in three dimensions and are sure we know what things look like. But what if there is a fourth spatial dimension that we don't know about? If we could move just one step in that direction things might be quite different. Maybe we shouldn't be so sure we know it all.

By the way... if you are wondering why those corners in the first picture line up so nicely, it's not a trick of where I was standing. It's math. X^2 - y^2 = -1 to be exact. Some of you will understand.

	See Fluid Geometry
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Helical

 
One of the interesting things I saw last Saturday on my walk was this white pine that has died and all the bark is stripped off. The wood has grown in a helix. I've seen this before, but this time I wondered why bad enough to go hunting for answers.

First of all, a helix is a smooth curve in three dimensional space. I like this animation from Wikipedia, which apparently won't display in blogger. So click the link and watch it. It's mesmerizing.

helix

You can see the two sine curves in two dimensions each on the YZ and XZ planes. Project them into XYZ space and there's your helix!

Have you ever watched squirrels chase each other around a tree in that pattern? Fun.

So the bottom line as to why... is... nobody is 100% sure.

It happens as a result of some kind of stress that causes uneven growth. There's some data to suggest that if there are strong prevailing winds, or if one side of the crown gets considerably more light it can occur.

It's NOT true that trees grow one way in the northern hemisphere and the other way in the southern. In fact, quite a few trees spiral one direction for a while and then change directions (so the deeper layers of wood will go the other way). They can even change more than once. Now that is weird! And that doesn't support the prevailing wind or sun theory.

Pinaceae (pine, cedar, fir, hemlock and spruce) are more likely to have the condition than are junipers or redwood. I've seen it on pines and cedar for sure.

If you want to read the best article I found about it, follow the link below to the Gymnosperm Database.

Then again, maybe it's the stress of the squirrel racetracks.

	See Why Do Trees Form Spiral Grain?
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Puppy Chow Math Test

 
Just a little project this morning to make some fun packages to take to people at work. At some point I realized I was doing several elementary school math story problems to complete the project. (and one a little more complex) I don't understand people who say they never use math. So here's a quiz for you. Answers at end.

I decided to make "puppy chow" snack mix for everyone.

Q1. How many people actually work in the mail room and on the presses?

Q2. Which is the better value? A 3-pack of red curly ribbon for 79¢ or a 6 pack of assorted colors for $1.35? The spools of ribbon are all the same size.

Q3. If the recipe calls for 8 oz each of rice and corn chex, but you want to add Cheerios too, and you are going to double it, and the rice chex is an 18 oz box, and the corn chex is a 14 oz box, and the Cheerios is a 16 oz box, how many boxes should you buy?



Q4. If the recipe calls for ½ c. peanut butter, but you are going to double it, and you have an 18 oz jar, how much should you use? (You melt the p.b. with oleo and chocolate chips.)



Q5. How big a pan or bowl will you need to mix it all?



Q6. The recipe calls for ½ c. powdered sugar, but you are doubling the recipe. How much powdered sugar will you need?

Sorry, no picture of the finished product, but here's the bag with sugar in it. You add the chocolate coated cereal and shake it up to give it a sugar coating too.



Q7. How many gift bags can you fill with what you just made?



Q8. With 6 spools of ribbon, each labeled as 10m long, if you use 2 arm lengths for each bow, will you have enough ribbon?

Q9. How many combinations of two colors can you make without repeating any of the combinations?

Q10. How much did this gift cost per person? The 3 boxes of cereal cost $2.99 each, and I had a $1.00 off coupon (not per box). The 2 bags of chocolate chips were $1.99 each. The peanut butter was $1.88 for the 18 oz jar and I had a 75¢ off coupon. The oleo was 88¢ a pound and I used 2 sticks. The sugar was $1.89 for 2 pounds, and I used about ¼ pound. The plastic bags cost $2.00 for 50 bags. The paper bags cost about a penny each (had on hand), and the ribbon was $1.35 for the six spools.

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ANSWERSA1. It would have been intelligent to ask someone, eh? O well, I decided to make a double batch and hope it would be enough. This is called estimating. I think this was an 8th grade lesson.

A2. 79/3= 26.3¢ each. 1.35/6= 22.5¢ so the 6 pack is the better value. Quick and dirty method: 79*2= $1.58 which is more than $1.35, so the 6 pack is cheaper. And more colors is nice anyway.

A3. One of each, because you are going to just dump it all in and not worry about measuring this stuff. This is not a fussy recipe, and you have a coupon for $1.00 off if you buy 2 boxes of cereal.

A4. You need ½*2 = 1 cup peanut butter. There are 8 oz in a cup, and half the jar of p.b. is 9 oz, so just use about half the jar. Please don't mess up a measuring cup to do this. This is not a fussy recipe.

A5. This answer requires you to know more about the density and bulk of the product, not the weight. Trial and error works as well as anything. The answer is- the two biggest pans you own, so there's room to mix the chocolate and coat the cereal.

A6. You thought this was going to be an easy one, right, as in ½*2= 1? Haha. The answer is, at least twice that much to properly coat the cereal. Don't believe everything you read in recipes.

A7. Just do it and see. Then please write the answer on the recipe card so you'll know next time you make this. You can make 18 bags with 5 oz of puppy chow in each one (yes I weighed them). That leaves about 3 oz that fell on the floor for the cook to eat (5 second rule). Just in case you are feeling OCD, it's interesting to note that 50 oz cereal + 9 oz peanut butter + 8 oz oleo + 24 oz chocolate chips + 4 oz powdered sugar = 95 oz of ingredients. 95/5= 19, which is almost exactly what I ended up with, allowing for some product loss that stuck to the pan, etc.

A8. A meter is about a yard. An arm length is about a yard. You have 6 spools * 10 m = 60 meters which equals about 60 yards. You have 18 bags to decorate. 18*2= 36 yards needed. You have plenty of ribbon.

A9. OK, this one isn't a grade school question, it's basic statistics. But the visual answer is in the picture of the bags. The formula is n!/(n-r)!(r)! where n is the number of objects to choose from (6 colors), and r is the number you want to use in each set, which is 2. The ! means multiply the number by every smaller number so n!= 6*5*4*3*2*1. (n-r)! = (6-2)!= 4*3*2*1, and r!= 2*1, so you end up with (6*5*4*3*2*1)/(4*3*2*1*2*1). You can cancel everything out except the 6*5 on top and a 2 on the bottom, so 30/2= 15 different combinations. But the visual answer is seen in the rows. The back row has red combined with every other possible color, so R-L we have red/silver, red/green, red/magenta, red/gold, red/blue. Then the next row forward is blue with every other possible color, so using the same order R-L we have blue/silver, blue/green, blue/magenta, blue/gold. This leaves blue/red, but we've already got that in the back row, so we only have 4 possibilities in this group (It's a whole different math problem if the order matters- in other words, if red/blue is not the same thing as blue/red). The next row forward is gold with every other possible color, so now we have gold/silver, gold/green, and gold/magenta, but that's all we can do that aren't duplicates. Next we do magenta/silver, and magenta/green. Only two new possibilities. Then in the front row is green/silver, and that's all we can do.

A10. Are you still here? Who cares? But quizzes have to have 10 questions, right? And I am curious. I snuck one little assumption in the question, and that is that you know there are 4 sticks of oleo in a pound. OK. {[($2.99*3)-1.00] + (2*1.99) + (1.88-.75)/2 + (.88/2) + [1.89/(2/¼)] + [(2.00/50)*18] + (18*.01) + [1.35*(36/60]}/18 Do each part of the formula {7.97 + 3.98 + .67 + .44 + .24 + .72 + .18 + .81}/18 Continuing 15.01/18 = $0.83 each. Not bad, not bad... everyone loved it. I got mega brownie points, and there was enough to go around, including for the guys running the presses, who often get overlooked when it comes to treats.

I suspect only two of my regular readers actually made it this far. You know who you are!

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Pi(e) is a Relationship

 
As noted the other day, there has been no blog proof of pumpkin pie. The mathematical proof of π is simple. In fact, it doesn't need any proof. It just is. It's the relationship of a circle's circumference to the diameter. It doesn't matter if the circle is large or small, if you divide the circumference by the diameter you get π. Exactly. Although there is no exactly, because it's an irrational number, a number that goes on infinitely with the division never coming out even. And it can't be represented exactly by a fraction. When we used 3.14 or 22/7 in high school, those are only approximations that are close enough for most problems.

If you want the area of a circle there's another relationship with π in it, πr².

Interestingly enough, we've just wandered into calculus. Since d=2r (diameter equals twice the radius), then the circumference = 2πr

If you integrate that "line" you get πr², and the integral of any line gives you the area under the line.

OK, I almost kept going, but I expect most readers have already had enough math fun for the day.

Pie is a relationship too. It's the relationship of fat, flour and liquid worked into a crust, and some filling, sweetener and thickening, baked to a state of perfection.



I'm not about to start complaining about sunshine, so you'll just have to deal with the dappled light in that picture. If you look to the far right, you'll see the filling in the blender. I guess that's the only not-quite-conclusive proof I can offer that it was made from scratch. The crust speaks for itself.



I've learned that a 10" pie holds twice as much filling as a 9" pie. How's that for an interesting relationship?



The relationship of the size of the pie to the number of servings is also an irrational number. No matter how big the pie is there is never enough and it doesn't come out even.

	See Rhubarb Pie See Crabapple Pie See Cherry Pie See Apple Cranberry Pie
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