LabBooks

My dream job is "tutor". I enjoy seeing "the light go on" when someone grasps a difficult concept, such as differentiation. 

I've heard people say that calculus is hard because, unlike arithmetic, it's not intuitive. I've even heard people grudgingly say that algebra and trigonometry are intuitive. But not calculus.

Things are intuitive when we are exposed to them so much that they become second nature. We aren't exposed to fractions - we are exposed to parts of objects and fractions are the way we are taught to think about parts. Why are fractions intuitive and derivatives aren't? Derivatives are the way we learn to mathematically handle change and we are surrounded by change.

Why is, say, multiplication, intuitive? You probably know how to multiply two big numbers using long, or partial product, multiplication. You multiply one long number by each digit of the other long number and then you add the products together, but you have to position each of them just right before you add them. Why do you do that and why should you be confident that such a complicated procedure will work every time?

Is that intuitive?

Did you know that all the arithmetic you use is based on a handful of assumptions that nobody tries to prove. One is: a=a. Everything is equal to itself. That might be true in a single case, but how do we know that it's always true? I'm not at all equal to the me of five years ago, but then, I wasn't the same person five years ago that I am today. This instant, I am equal to myself.

Can you divide and always come up with whole number answers. There's a perfectly legitimate and useful way to do that and you might not remember it, but I can just about guarantee that you did it in elementary school! 

How do you know that 2+2=4, and why would you think that it is always the case? Can you prove it? We take an awful lot for granted.

Isaac Asimov was a great popularizer. Through most of his publication history, word processors didn't exist. Have you ever used a typewriter? Typewriters were what we used to create documents when I was in college. Word processors came out while I was in college. The typewriter word processors let you look at sentences you typed before you committed them to paper, but the computer programs were really cool. You could type an entire book, then go back and make corrections, change formats, and even add pictures (!!) before you printed it out. And then there were desktop publisher applications that made it all much easier and added a lot of options.

But the end result was still what I call "flat copy". The page just sat there while you read it. I still use word processors, for instance, I am typing this blog on a word processors app, Google Docs, on my cellphone. While you are reading it, it just sits there. I have embedded videos into some of the blogs, but they're still not anything you could call "interactive".

What I really enjoy using for educational materials is a spreadsheet application.

There's a link up there to the right that will take you to the download page of my other website. The page is called "Excursions". Most of the free downloads there are programs (like the statistics spreadsheet DANSYS) and their user guides, and LabBooks.

LabBooks are textbooks that are spreadsheet documents. Since they are spreadsheets, they're not flat copy. While you're reading them you can be doing other things, too.

LabBooks are lifelong projects for me. I might not live long enough to finish one, but I place them on my Excursions page when I update them. I just reposted the Mathematics LabBook. I waited until I had completed the first part of the first section. It's about the natural numbers (AKA the whole numbers) and the basis of arithmetic. All those questions I asked above? Read the Mathematics LabBook and you will understand the answers.

It has exercises you can do on the page and some buttons you can push to generate problems and get the answers. And you can do your own calculations in it.

I like to open up a concept and show how the insides work.

There are a few loose ends I need to tie up in the rest of the first section. For instance, I've been saying that I will show you how to memorize long numbers in mental calculations, and I will do that on the next sheet.

Talking about interactive documents, I would think the next wave of educational software might be virtual reality. A housemate is into VR. It makes me dizzy but I can imagine "Mister Wizard in a can." 

Vectors

This first paragraph... you're either going to know what I'm talking about or you won't have the slightest idea. I didn't study tensors in college (or anywhere else except on my own time) and I suspect that they're still a graduate level subject. But they're not that complicated. Mathematicians use collections of values that can be handled as a single entity. They're called "matrices". A tensor is basically a multidimensional matrix of measurements. A zero order tensor is a scalar. A first order tensor is a vector. A second order tensor is a two dimensional matrix. A third order tensor is a three dimensional matrix, and so on.

The important thing about these things is that they are collections of many values but they can be handled as a single unit. It's like...you can buy a dozen eggs at the grocery store and bring them home separately - that's 24 trips to and from the store. Or you can put them into a carton and put the carton in a bag, and bring them home all at once. Just forget "eggs" and think "carton" until you get ready to make omelettes.

Tensors are advanced math and people generally think of advanced math as a kind of puzzle for folks that like that sort of thing. Well, abstract math is (sort of). But tensors are not abstract math - they're the other kind - practical math. In school, you study practical math. For most of us it goes like this: elementary arithmetic, algebra, geometry, trigonometry and precalculus (or college algebra), and then, in a college science curriculum there are differential calculus, integral calculus, and maaaaybe differential equations.

None of that involves tensors. You'll probably study some linear algebra ( that's matrices) and, in physics, you'll learn about vectors. But there's a lot (!) Beyond that….statistics, calculus of variations, numerical analysis, discrete mathematics…

The point is that all this stuff isn't abstract nonsense. It's all...and here's the kicker...it's all quick and easy labor saving devices. That's right. Advanced mathematics is there to make life easier for people that need to do certain jobs.

Take matrices from example. Let's say you had to figure out three values and you have three equations that contain them. Say, you overheard three people talking. One said, "I have 25, 7, and 43, so I have 490." Another said, "I have 13, 9, and 17, so I have 228." The third said, "I have 3, 23, and 37, so I have 488." You might reasonably think that they're talking about numbers of things with three different values, and you guess that they're naming the things in increasing value so you set up the following equations:

25X1+7X2+43X3=490

14X1+9X3+17X3=228

3X1+23X2+37X3=488

There are several ways to solve for the values of the three variables that make them all true at the same time. One is called elimination and it looks like this.

I counted about 27 (grueling) operations there. Here's the "advanced math" matrix method.

See, you can treat matrices like individual numbers so, once I had the coefficients of the variables packed safely away into one matrix (A) and the numbers on the right side of the equations packed into another (B), I just inverted A and multiplied it by B, two easy operations on the spreadsheet, and I had my answers.

It's pretty clear that the three were talking about money - pennies, nickels, and dimes.

It's a lot easier working with matrices than with individual numbers.

By the way, scalars are just individual numbers. A vector is a row (or column) of numbers. A two dimensional matrix has rows and columns, like the ones I used above. You can have a stack of two dimensional matrices to form a three dimensional matrix, and you can keep going adding more and more dimensions until your brain explodes.

When you're talking about tensors, you're usually talking about measured values and things can get pretty deep, but I won't here.

Many of the values that physicists work with have two parts, so they pack well into two valued vectors. Think back to all the things I measured in the playground.

I started at the trailhead and walked to the playground measuring the distance using AllTrails. It was 0.3 mile. That would not be enough for a physicist, though. They would also want to know the direction. My direction was almost due west or pi radians from an east-west line. The vector would be (0.3,3.15). It could also be represented by an arrow pointing west with a scaled length representing 0.3 miles. The vector would represent my displacement. Distance is a scalar; displacement is a vector.

The funny thing about displacement is that it's the distance traveled from start to finish. On a loop hike, displacement would be exactly 0, since, all told, I would have gone nowhere.

Speed is also a scalar. I walked about 1.8 mph. But physicists talk about velocity, which is a vector consisting of speed and direction. I walked 1.8 mph west.

Acceleration is also a vector consisting of change of speed and change of direction. When I was spinning the phone on the cord, I tried to keep the speed constant but there was still acceleration because the direction of the motion was constantly changing. (Actually, since I didn't do a very good job keeping the speed constant, both were changing.)

Most of the vectors in "undergraduate" physics are two-valued. More advanced physics and engineering get to use larger vectors because they're dealing with three dimensions (space) and four dimensions (relativistic space-time). But our universe is growing and some physicists think that we need ten, eleven, or, maybe, an infinite number of dimensions to describe it.

Are big vectors a problem? Well, they're hard to visualize but statisticians, social scientists, researchers, economists...they've had to deal with big vectors for a long time because every case in a dataset is a vector that might be described by two, fifteen, or thousands of values. Just think about what you look like in a census report: age, ethnicity, residence, number of people in your family, whether you're the head of your family - all values in a huge vector.

We will be talking about other quantities in the future: force, work, energy, magnetic fields. If you've never seen vectors in action, you'll get to see how they work.

Meanwhile...

Obituaries are a thing.

America is an odd culture. We avoid the topic of death like a plague yet the Internet is packed with discussions of people who read the newspapers (and Internet) obituaries everyday of their lives and why they do it. A cursory examination of Google Scholar did not turn up any studies, nor could I find any data, on people who read obituaries.

There are many citations on Google Scholar about obituaries, though...how to read them, how to write them, how and why to research them, what they tell us about the people they discuss, what they tell us about society in general.

It's interesting reading. You might take some days to study obituaries.

Why am I talking about them? 

Well, it's a lead in to my excuse for my absence from this blog for a couple of months.

First, a very close friend died. Learning can be…somber, but it's always potentially edifying. I learned that, as you get older and old friends die off, it gets easier to shrug it off. Bereavement is about loss, not pity for the deceased. After you loose enough, you get used to it and life teaches you that everything isn't necessarily about us. The world goes on.

When you know you are dying, you can either say, "I'm dying," or "I'm going to live until I die." Mike showed that it is entirely possible to do the latter with all the determination of a 20-something with nothing but life ahead of them. Another person I knew who "died well" was my father. You can learn a lot from folks like that.

A week later, I went to church and couldn't sing. The next day, my annual episode of bronchitis was in full swing. I have learned to expect it every year. Some truths are just nasty. So, for a month, I sat around hacking my brains out and wearing myself down.

A close friend and college math professor was scheduled to attend a mathematics conference in town and I had said that I would go with him. It was the week after my hacking cough had let up and I was flat worn out but I keep my word, and I really did want to go since I'm an amateur mathematician and it would give me something to blog about. I had the opportunity to go as his guest, which meant a 95% discount. On my pension, it was an offer I could not refuse.

Of course, it was mostly sitting through lectures and strolling around math art exhibits and vendors (I bought gifts for folks). There was a surprising lot about my own, not so deep interests - math education and statistics. So, at least I enjoyed further abrading my life force. And I got some close-ups of the big blue bear.

At least, between bear and horse, the more personable one is the more accessible.

Finally, I'm on the mend and I've even knocked out another terminal hike (that leaves three to go) and that will be the subject of my next blog.

I recommend not avoiding mental adventures like community lectures and conferences because you think they will be boring. It's all mindset. Things that you expect to be boring usually are.

Shake out the obituary section of a newspaper. Look at some obits online. Do they have a consistent structure? Do you know an obituary reader? If so, why do they enjoy reading obituaries? What do obituaries tell us about our attitudes toward death? What do obituaries tell us about our attitudes toward life?

Calculus in your life

Mathematics is a language that is ideal for describing how the world works and building models that help understand the underlying processes and make predictions.

As a tutor, I heard a lot of, "I'll never use this. Why do I have to learn it?" and, honestly, I sympathized. I was helping high school students learn concepts that I was taught in college courses. Some intended to go on into technical or theoretical curricula in universities...but not all.

So, why advanced math in high school? Well, first, calculus, as difficult as it's made out to be, is neither advanced nor difficult. Once you understand it, it's no more difficult than any other math. It's a language. It has a vocabulary and it has rules for putting it's "words" together to make sense. The problem is that many calculus teachers don't understand it, at least not well enough to impart understanding to their students.

Further, calculus is the last tool you need to understand the world most people see in everyday life. The sequence goes like this:

Arithmetic gives you the tools to count and measure.

Algebra helps you solve problems in math.

Trigonometry let's you figure out the distance across a river without actually having to get your feet wet - that is, surveying.

Geometry is what you need to design structures that will stand up.

And, calculus...calculus is the tool you use to deal with change, because the two concepts

(only two) at the center of calculus, differentiation and integration, are what we have to deal with rates.

You see, a derivative (that's what you find when you differentiate) is just a slope. Every handyman knows what that is - it's rise over run. Measure a horizontal distance, that's the run, and then measure vertically up to the ramp,  or stairs, or roof, and that's the rise. Divide the rise by the run and that's the slope. Where it gets a little complicated is when the object you're measuring the slope of isn't a straight line. What if it's the trajectory of a bullet or a curved pipe?

The speed of a car, or any other rate, is a slope. The run is the time it takes for the car to go a number of miles. The number of miles is the rise. Rise over run….miles over hours….miles per hour. But cars rarely travel at a constant rate, so it's sometimes nice to know how fast a car is going at any particular time -that's called "instantaneous speed"and you need calculus to figure that out...or a speedometer, but a speedometer is an analog calculus calculator. It adds tiny chunks of speed (of the wheels) to come up with an estimate of current speed. That's the other concept of calculus….integration.

Integration adds tiny pieces of area, volume, rates, what have you, together to get a total. It does the impossible by adding together an infinite number of infinitesimally tiny chunks to get an infinitely accurate total.

For instance, if you knew exactly what the instantaneous velocity of a car is at every instant of a trip, you could add all those rises together to know exactly where the car is at any particular point in the trip.

But does anyone really add infinite quantities? That would take an infinite amount of time. Eh...no. But calculus provided ways of estimating integrals to any degree is accuracy. Those tools are called "numerical analysis".

Want to actually do the addition? Luckily, there is a way, because the integral is the reverse process of the derivative! If you know a derivative, you can just start there and go backwards to find the integral!

Some wag once said, "there are two things in calculus, differentiation and integration. Everything else is application."

So what do I think that everyone should know about calculus? I think they should know what it is and what it's good for. They should also know where to pick up the knowledge of how to use it in case they encounter a problem that requires it.

Carpenters, construction workers, and practical statisticians rarely, if ever use calculus, but the mathematical tools they use are based on calculus and someone had to work those out. If you want to make a career out of fiddling with numbers (that's what theoretical mathematicians and scientists do) or if you want to build the mathematical tools that other people use, then expect lots of calculus. And if you plan to follow me into next year, also expect some calculus.

I'm about to embark on an adventure into the hard sciences, astronomy and physics, and calculus is a big part of their language but, hopefully I can explain well enough for you to catch the passion for lifelong learning while I break open the hood and show you all the workings inside the world.

By the way, if you feel adventurous and want to study calculus or any other advanced math, MIT has some great online courses at https://www.google.com/url?sa=t&source=web&rct=j&url=https://ocw.mit.edu/index.htm&ved=2ahUKEwjZ2MfQiuDkAhXMuZ4KHXNiDiwQFjAAegQIBhAD&usg=AOvVaw0l1lsPpezEpLxudzjIP0u4

--- How big is Creation Rock 2 ---

Frankly, I have no idea.

I've seen two websites that have the same statement, that both Shiprock and Creation Rock are 300 feet tall...but they're different heights. I've also seen that they're both taller than Niagara Falls.

Actually, it's hard to figure out which rock is Shiprock and which is Creation Rock, as this article at the Red Rocks Park website explains.

http://www.redrockspark.com/history/history-of-red-rocks-park

And obviously my survey wasn't correct, though I'm not as bothered about that as some might expect.

Success is an American ideal. Researchers rarely publish reports on failed experiments despite most books on research design encourages researchers to do so.

You often learn more from a well designed failure than you do from a success. A success gives you an answer. A failure gives you a bunch of interesting questions, such as, "What went wrong? Was my study design flawed?" "I was expecting this to work, it should have worked, why didn't it?"

In 1887, Albert Michelson and Edward Morley wanted to study the "stuff" that light waves flowed through. They set up and experiment that shot a stream of light in two different directions, one at a right angle to the other. They wanted to measure the difference in the speed of light through the paths. They were sorely disappointed when there was no difference.

The problem was that light doesn't propagate through "stuff" like sound or water waves, and this failure started a line of study that lead quickly to Einstein and special relativity.

So, I learned some things from my "failure" and I'll let you in on those things. And, one of the great things about adventures vs. studies is that adventures have a focus surrounded by lots of other "stuff".

I knew that Red Rocks had a concert planned that night (summer's like that at Red Rocks) and that I needed to do what I was going to do before 2:00 afternoon, so I hired a Lyft and headed straight there. The driver let me out at the Trading Post Trail trailhead.

I was delighted by the diversity of plant life on the trail.

                                                   [White Anemones and Golden Asters]

I was rather concerned by the many signs that urged visitors to stay on the trail. I certainly understood the reason - there are delicate plants and animals that live at Red Rocks Park and they don't respond well to crowds tromping all over them, but that was going to interfere with me finding a site for my survey.

As for the plants in the above photo, the anemones are easy to identify by their simple (but often brightly colored), five petaled flowers and deeply lobed leaves. The asters are a little more difficult but many similar flowers have more irregular petals and/or broader central disks.

In contrasts to the tiny flowers are the gargantuan rocks in the park. Red Rocks is justifiably famous for the huge natural amphitheater, home to internationally attended and broadcast performances, but the park itself is a geological wonder.

                                                                        [Big rock]

As the Pacific Plate crashed up against the North American continent, pushing the Colorado Plateau up to form what would become the Rocky Mountains, land around the edges were severely twisted and buckled. These layers of sediments are often tilted to right angles to the rock to the east and are sometimes turned over completely. There's evidently some debate as to whether the rock is sedimentary sandstone or gneiss. In support of the frustrated geologist that snorted and corrected me when I called the Red Rocks "sandstone", rocks this twisted are usually considered metamorphic and, therefore, gneiss. Regardless, the red is prominent and is caused by the same thing that makes old iron red.....rust.

                                                                 [Slanted rocks]

In the photos above, the big rock center stage is my target, Creation Rock. It's the largest of the three massive rocks that make up the amphitheater.

Although Red Rocks Park is the most famous display of these tilted Goliaths (and, in my estimation, the best), there are certainly others. Technically, they are called the Fountain Formation. They stretch through Wyoming and Colorado. These photos bear a strong resemblance to the Flatirons just outside Boulder, and for good reason since the Flatirons are another example of the Fountain formation. Other famous outcroppings are the Garden of the Gods in Colorado Springs. Other instances are the Red Rocks of  Eben G. Fine Park in Boulder, Roxborough State Park, and Red Rocks Canyon in Colorado Springs.

Here are some tiny blue larkspurs.

                                                                 [Blue Larkspurs]

Just behind Red Rocks is Mount Morrison and to the south is Mount Falcon.

                                                      [Mount Falcon from Red Rocks]

I was considering hiking up Falcon when I finished at Red Rocks but the brutal heat of the Colorado summer convinced me that I would be miserable and decided to hold off until cooler weather.

The Fountain Formation marches off to the south between the Front Range and the Hogback ridge to the east, From there to the Mississippi River is grassland plains.

                                                  [Slanted rocks to the south of Red Rocks]

                                               [Wavy rocks on the Trading Post Trail]

I overheard a lady telling a group of children that these waves frozen in stone were the result of erosion from the action of water. My understanding is that the rock carvings of Red Rocks Park and other parts of the Fountain Formation are mostly due to wind erosion. A little further to the north, on Dinosaur Ridge, I over heard a tour guide tell a completely different story.

As incredible as it may be, Colorado was, at one time, covered by a shallow sea. All the dinosaur activity in the area were huge beasts sunbathing at the beach (or maybe a marsh). This was before the Colorado Plateau was uplifted. So, these waves, frozen in rock were not due to erosion, but deposition - sand deposited by the gentle wave action of the inland sea, then covered by more sediment, packed down, and turned into stone under great pressure until they were uncovered millions of years later by the wind. And if you've been in any of the downslope winds in this area, that won't be a great stretch of your credibility.

                                                                 [Jumping cholla]

This is another good reason to stay on the paths at Red Rocks. The Jumping Cholla is notorious for reaching out and touching people. Still, these (cactuses) are still my favorite wildflowers and Denver area, being desert, is a natural habitat for them.

At the end of the Trading Post Trail is, of course, the Trading Post, which is also a museum of pop music. It is located to showcase some of the big rock formations, such as this huge column.

                                                                 [Big rock column]

Many of the formations are named, but there are so many....I can't find a name for this one.

I looked around the Trading Post and asked a couple of staff some questions verifying that I was going to have problems finding a place to stretch my 100 foot cord. I finally settled on a line of sight near the Trading Post for my two locations. The first is at a fire plug across from the building. Here is Creation Rock from that site. The road branching off to the left is Shiprock Road.

                                                       [Creation Rock from Site H]

The other site was a little ways up Shiprock Road to the south. I could just sight the summit of Creation Rock above the boulders to the west of the road.

                                                          [Creation Rock from Site G]

Since I knew that I would have problems with this data, I started looking for ways to double check my work, multiple ways of estimating bearings and distances. I measured the distance between my two sites using an app on my phone called Altitude (created by PyGDroid and available on the Google Play Store). The app calculates distance traveled using GPS. I found the distance between the two sites to be 0.05 miles. That converted to 264 feet and, later checking Google maps for a distance, I found it to be about 257 feet, which is the distance I ended up using.

Here is a shot from the Trading Post back down through Bear Creek Canyon where Morrison is situated. The lakes in the distance  are the Soda Lakes at Bear Creek Lakes State Park. I'll get to that later.

                                                    [Bear Creek Canyon from Red Rocks]

My altimeter reading told me that Site G was about 40 feet above Site H and I could look down on top of the tall Trading Post building from Site G, so I felt pretty confident with the phone app.

I went to work getting bearing readings from my surveyor's compass and I checked them to the nice north to south bearing of the Front Range. My bearing from Site G to Creation Rock was 313 degrees and, since I was east of the summit, that made sense. The north to south line is 360 degrees and everything to the west would be less than that. Sighting back to Site H, I found a bearing of 72 degrees. Since it was east of the north - south line, it should have been between 0 degrees and 90 degrees, and it was. I could add the difference between the bearing to Creation Rock and the north - south line (47 degrees) to the 72 degrees west to find the angle G, which was 119 degrees.

The bearing from the fire plug to Creation Rock was 257 degrees and the bearing back to Site G was 290 degrees - both were west of the north - south line, so those made sense and I could subtract them to find angle H, which was 33 degrees. That felt right. The angle of inclination from G to the summit of Creation Rock is 16 degrees and that from H to the top is 15 degrees, so I had the distance of the base of my triangle (side e) and the angles G and H, and the angles of inclination. That was what I needed. Here's my trigonometric plan diagram again.

The problem was that I had no idea what the elevation difference was between me and the base of Creation Rock. I was going to calculate the height of Creation Rock above me, not above its base. I didn't expect to be close. Well, I went with it and started back down Trading Post Trail to Morrison.

                                                     (From Red Rocks to the south]

On the way, I saw this patch of prickly poppies. That was the first western wildflower to confuse me after moving from Alabama. It looks like a thistle until it blooms and, then, that's certainly not a thistle bloom.

                                                                 [Prickly poppies]

And there was this paintbrush flower.

                                                                [Paintbrush plant]

And this rose.

                                                                             [Rose]

I didn't know that roses grew wild in Colorado.

Back on the road, I saw these huge sweet pea plants and milkweeds

                                                                    [Sweet peas]

                                                                    [Milkweeds]

I've mentioned that milkweed is one of my favorite wildflowers. I think that it grows over most of North America. I would occasionally see it in the South but it seems to like cooler climates. The further north I would travel, the more milkweed I would see.

We had a few stalks trying to grow in our back yard but one of the recent storms took them out. I was a little disappointed, there was a big flower head developing on one of them, and they are the only plant that Monarch butterflies will lay their eggs on, but then I saw all these plants in Morrison and felt better about it. I can enjoy other people's milkweed.

After a milkshake at the Mill Street Eats, I headed down Bear Creek Trail. It's been awhile and, after all, this is the Bear Creek Commentaries. I felt the need to reconnect. I also wanted to get a shot of Red Rocks Amphitheater from Mount Carbon to put it in perspective.

                                                                [Mount Carbon]

I found Bear Creek Lakes Park just as I left it, a blazing expanse of open space. By the time I reached the top, I was ready for a snack, water, and a long rest.

                                                       [Denver from Mount Carbon]

Here's Red Rocks with a magnification of X2. It may not seem so big but compare it with the prominence of Mount Morrison behind it (741 feet) and consider that this photo was taken from 3.9 miles away.

                                                       [Red Rocks from Mount Carbon]

My preferred route up or down the eastern flank of Mount Carbon is the Mount Carbon Loop Trail, which has great views of Denver and Bear Creek Valley. It also has a diversity of wildflowers such as this prickly pear cactus, certainly not a rare plant, but, like all cactuses, has an intricate, silky, showy blossom.

                                                               [Prickly Pear Cactus]
By the time I walked around the Fox Hollow Golf Course and reconnected with Bear Creek, I was whipped and spent some time on the shoulder of the trail. I would have used the bench across from me but Colorado, evidently, does not believe in shade and most park benches are situated to catch full sun. (what is that?)

Still, it allowed me to catch this bug's eye view of the grasses that line the trail.

                                                                       [Grasses]

I made it to Wadsworth Boulevard and chugged down two cans of Arizona Green Tea before calling a Lyft taxi to return home.

Now....the calculations.

Here's the geometry again.

 I know that angle G is 119 degrees, angle H is 33 degrees, and the length of side e is 257 feet. Angle E is easy because the sum of the angles in any triangle add to 180 degrees, so 180-(119+33)=28 degrees. That makes sense. Angle E is the angle between the two sites as seen from the top of Creation Rock. They are only separated by 257 feet so the angle of separation should be sorta small.

Since I have all three angles in the triangle EGH and the length of the base side, I can now use the law of sines to figure out the other two sides.

                                                                      [Law of Sines]

The length of side g, the distance from site H up to the top of Creation Rock, works out to be about 479 feet. The length of side h is about 298 feet.

Now, the diagram is a little misleading here. The point F looks like it's on the line GH, but it's not - it's way back behind the line. It is the point directly beneath the summit of Creation Rock. I calculate the length of line EF by either looking at the right triangle EGF or the right triangle EHF, and I will do both because the difference I come up with is important.

Right triangles are easy. In this case, I have the long side, the hypotenuse of the triangles (sides g and h) and the angles of inclination at G and H, and I want the length of the opposite sides. That's actually the measures that I want because it's the height of Creation Rock from the summit down to the level of my surveying sites. In other words, if I find the sine of one of the angles of inclination from the site up to the top of Creation Rock and multiply that by the distance from the site to the top, I'll have the height. I use my spreadsheet to calculate the sines and I come up with 82 feet (from site G) and 124 feet (from site H).

These results are certainly far short of the actual height of Creation Rock, but do they make sense?

Well, the difference between the two calculated heights is 42 feet. That's the difference in altitude between Site G and Site H. To walk from the Trading Post to the fire plug at site H, I had to walk slightly down the hill, then to walk to site G I had to walk around and above the tall Trading Post building. I could look down onto the roof. 42 feet works.

Also, I pulled a topographical map from the Internet site:

https://www.topozone.com/colorado/jefferson-co/locale/red-rocks-amphitheater

The contours are 40 feet elevation apart.The contour on level with the Trading Post seems to be about 3 contours above the base of Creation Rock, so that would explain the discrepancy between my measurements and the reported 300 foot height of Creation Rock. It looks like site G was about 3 contours, or 120 feet above the base. Even with that, I was off, so, no cigar.

Most likely, distance e was my most inaccurate measurement. The compass bearings seemed to agree well with the line described by the mountains.

But it was a fun and scenic adventure, so I'm happy.

The methods I used have served me well in the past. This trip was plagued by several unexpected problems - difficulty in finding two good sites being a major one. You may be able to use trigonometry in your own projects. One suggestion is to avoid angles that are too steep (within between 70 and 110 degrees) or too flat (between 340 and 20 degrees.) At these angles, a tiny difference in slant can make a huge difference in the value of a trigonometric function, or vice versa, so that can throw your precision off considerably.

There are a lot (!) of resources on the Internet about trigonometric functions and surveying methods. I can recommend CK12's Trigonometry textbook, 2nd edition by Lori Jordan, Larry Ottman, Brenda Meery, Art Fortgang, Andrea Hayes, and Mara Landers  and you may want to look at their other offerings at:

https://www.ck12.org/student

By the way, according to the topographic map, Creation Rock is either 280 feet or 400 feet tall according to where you consider the base to be.

--- Mathematics in the colleges ---

Just looking at the websites of the colleges and universities in the area, it's hard to see any real focus in their mathematics departments. They seem to have a broad range of interests. I think I see a strong leaning toward discrete mathematics (combinatorial mathematics, graph theory, fields that deal specifically with whole numbers) and theoretical math at the Denver University. There are a few faculty members that share my interests in educational mathematics and statistical tools.

Statistics seems to be a big topic in the Colorado School of Mines. They make their areas of research more explicit on their website. There seems to be a lot going on with wave phenomenon, spatial statistics, computational math and analysis, and mathematical biology. For a school that focuses on applied technology, those topics don't surprise me much.

I walked up to the Denver University campus last Friday to sit in on a presentation and subsequent reception. It was a nice day and I got some nice pictures of the school of mathematics and science, which is spread across several buildings along Illif Avenue.

The water garden nearest the Newman Center, one of three on campus, is named Dan's Garden after Daniel Ritchie, a major donor to the university. It is in front of the Daniel Felix Ritchie School for Engineering and Computer Science, which is named after his father.

                         [Dan's Garden and the School for Engineering and Computer Science]

The three buildings to the west are all named for the philanthropic Boettcher family - the East and West Boettcher Buildings and the Boettcher Auditorium. Charles Boettcher was born in Kolleda, Germany in 1852 and his son, Claude, established the Boettcher Foundation in 1937, which supports education and research in Colorado. The park between the East and West Boettcher Buildings has several sculpture including busts of Charles, Claude, and Charles II Boettcher.

                                                     [The Boettcher Buildings and park]

The other building that houses the School of Natural Sciences and Mathematics is the Clarence M. Knudson building.

                                                                   [Knudson Hall]

The Robert and Judi Newman Center for the Performing Arts stands to the east. This cyclopean building sports a sundial on the south wall and, not a rose window, but an embossed rose, on the western facade.

                                                                 [Newman Center]

After wandering around this southern strip of the campus, I headed to Knudson Hall and the lecture. It was presented by Florencia Orosz Hunziker of Yale and was entitled "Fusion Rules for L(25,0)".

I enjoyed the lecture and... in truth, I didn't understand a single word she said. Actually, I did understand two words. I knew that there is something called a "tensor" and I understand "isomorphism". It was way above my head, but that was pretty much what I expected.

So, why did I go? First, I wanted to meet the mathematics faculty and, second, I wanted to push myself beyond my boundaries. Although I didn't understand the principles that Ms. Hunziker presented, I found that I could follow her thoughts and I recognized the method that she and her colleagues were using in their research. Afterward, in the reception, i had a brief conversation with her and was gratified when she didn't look at me as though she thought I was an idiot.

Actually, the all the attendees were friendly and very human. The conversations after the lecture were not at all above my head.

And as a bonus, the lecture rang bells.

I've dipped into theoretical mathematics a few times but it tends to be about three graduate courses further than I've been but I recognized a few things. Ms. Hunziker talked about Virasoro algebras. I had never been exposed to Virasoro algebras before but she kept drawing triangles on the board representing three-way isomorphism. Not really enjoying the feeling of ignorance, I looked up the topic at home. I can't say I understood much better but I noticed a recurring connection with particle physics and string theory.

One of the much cherished principles of physics has been supersymmetry, a belief hat there has to be an underlying, unifying pattern in the universe. From what I have gathered, supersymmetry has been rather disappointing. For instance, matching subatomic particles have failed to appear where they were expected. This pattern they that have been searching for seems to be a two- or four-fold pattern - sort of, "everything has a mirror image".

But I remembered reading a book in school by Sir Arthur Eddington, "New Pathways in Science". One of the chapters discussed Dirac's "Theory of Groups" and his idea that the universe is governed by pentads of operators. I don't see much about pentads anymore. It looks like it was one of Dirac's ideas that didn't really catch on. I wonder if the three-way operators are one side of a pentad.

So although I still don't have the foggiest, I did enjoy the seminar.

I've noticed that most progress comes from stretching beyond your limits. My memory has never been great but in memory tasks, I could never move forward until I put down the text and pushed myself to recite. I couldn't use new problem solving techniques until I moved past the examples into the exercises.

I once read a book on differential equations and came away as ignorant as when I started. Then, about a year later, I read the book again and, magically, I understood everything. I say, "magically" because I had done nothing mathematical in the interim. Obviously, my subconscious had been working in the meantime.

Adventuring is moving out into uncharted territory. The excitement of adventuring is the novelty.

In most cities and towns, many of the streets, buildings, and parks are named after people - people that, at least at one time, were important to the history of that place. What are the place names in your area and where do they come from?