Photos during the time of ...

(With apologies to Mr. Gabriel Garcia Marquez)

Avoiding everything is getting old and is certainly cramping my style, but slowing the transmission of a novel virus has its points.

I've mostly been wandering around close to home. People are out socially distancing. It is certainly a time to be out observing social behavior under atypical circumstances.

How are people behaving around you? How are you reacting to the different situations?

I've been back to Fiddler's Green to see more of the statues and I walked down Little Dry Creek Trail to look at Holly Reservoir. Here are some photos.

A reminder of the old west in modern Arapahoe County.

That odd little mound near Arapahoe Station.

An ensemble at Plaza Tower One, Village Center. A bear, plates falling down and some logs.

The elephant puzzled me until I saw the mouse on the pavement in front of it.

Denver likes murals.

This wild boar looks like a matching statue over near Englewood Station. Much of the art in this area is also part of the Denver Museum of Open Air Art. 

This piece of modern art...well, I'll let the artist explain it….

Giraffes!

They're still working on Marjorie Park.

There are extravagant water features all around Denver. This one is at the apartments called "The Cascades". I suspect there's a little nose twerking of nature here in the high desert.

Pike's peak from Quebec.

Another consequence of the virus.

A photo I took of the sun through a welder filter. You know that photographing the sun directly will damage your digital camera, right?

Interesting tunnel under Arapahoe on Little Dry Creek Trail. It's blocked now but maybe later...

Little Dry Creek at Holly Park

Mount Evans from Holly Park. They're done nice views from this little park on Little Dry Creek Trail.

Holly Park

Although Little Dry Creek is almost never dry, Holly Reservoir usually is. Like many stream constructions in the area, Holly Reservoir is a buffer in case of flash floods.

If you want to hike the whole thing, Little Dry Creek Trail begins at Yosemite near Briarwood and Davies Streets and runs about 4 miles to the Highline Canal. It's a well maintained trail, easy at sections but the stretch along Arapahoe is a constant grade that can wear you out after a time. It grades up toward the east.

One of the socially acceptable activities during the time of The Virus is hiking. If you're into biking, that's okay, too.

Terminus: Ridgegate

Ridgegate station is the southern terminal for three lines of the light rail system in Denver: the E, F, R lines. It's just about as far south as you can go and be in the Denver area. Just south of there are the bluffs, a wild grassland.

I hear that there are companies wanting to build there, but there are also concerned citizens fighting them to keep that area wild.

Ridgegate is growing. The light rail extension from Lincoln Avenue to Ridgegate Parkway is new and I haven't been south of Lincoln on it. I visited someone at Sky Ridge Hospital near Ridgegate a couple of months ago.

Currently the terminus is isolated but in easy walking distance of the Ridgegate community, it's shops, and the hospital. My destination was The Bluffs Regional Park.

Like all train and bus stations in the Denver area, this one has been adorned by a known artist, in this case, two artists. The names are Erik Carlson and Erica Carpenter. Decidedly modern, the art incorporates local ranch brands and technological symbols into an exhibit called "End of the Line." I'll let them tell you.

Here's some of the art.

The station is large but sort of lonely out on the plains.

As near as it is to a dense residential area, it's separated by I-25. A short hike brought me there. The entrance to the town is guarded by a brass elk at Cabela's parking lot (I guess it's an elk.)

Long, long ago, in a land pretty darn close by (I live in a suburb of Denver), the Pacific Plate crashed into North America and buckled the continent forming the Colorado Plateau. Except for a few volcanoes that belched a lot of ash into the wind, it was a fairly well rounded plateau, but rains fell and, soon rivers formed (like they do) and the South Platte and Arkansas Rivers started carving out deep gorges and depositing silt and rock east of what would be the Rocky Mountains. Mixed with the mud that formed when the volcanic ash decomposed, there was a deep blanket of gunk sloping out across the plains. To the south of what is today Denver, quartz rich sands got washed down and formed a hard layer over the softer stuff.

Time and rain kept flowing and water flowed both west and east. Tributaries of the South Platte and Arkansas Rivers kept excavating. To the north of Denver, nature wore the land down to a nub, and today we have the gently rolling hills of the western plains. To the south, that hard capstone resisted erosion in places and now we have the mesas and buttes like Castle Rock. Between, the ridge of Palmer's Divide split falling waters apart to run into the South Platte to the north, and the Arkansas to the south. And right on the northern border of Palmer's Divide is this 

I'm sure the little Creek that flows through Lone Tree had something to do with forming the hills in Bluffs Regional Park but they look like wind might have played a big part, too.

The Bluffs are an island of grassland above the surrounding residential areas. The wind was strong and cold and I'm sure there was plenty of wildlife out of sight. Still wintery weather in the Denver area, it's too early for wildflowers to be blooming. The main draw today was the views of the mountains. Only four miles further south from where I live, the bluffs make Pike's Peak seem far closer.

Pikes Peak

Mount Evans and the Front Range

Devil's Head

At the overlook, there's a little circle of stones that serve as a place to sit. It served me as a place to eat my lunch.

A stone disk set in the center of the circle points out some of the surrounding mountains.

After a short hike, I headed back down to the station and took a train back home.

Do you know the geological history of your area? There are surprised there. Check it out and then keep your eyes open for clues to it's past as you hike.

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...

I hope you don't think that the only things I do are what I report in my blog two or three times a month. Most of my activity is family and community business. I have a Carolina dog that keeps me busy at home and along with washing dishes and cooking occasionally, I make frequent trips to local grocery stores and there is the weekly trip to the old neighborhood where I am still the assistant librarian at the church I used to attend.

But I'm supposed to walk at least 20 minutes every day to keep my heart going. Almost literally, "if I stop, I stop." Luckily, even my local walks are interesting. Some of the neighbors like to talk and most of them have friendly dogs (Denver is dog territory). The scenery is even spectacular in the neighborhoods.

A while back, I walked over to the other side of Arapahoe to check out Fiddler's Green. It's at the north rim of the valley formed by Little Dry Creek. It's the southern border of the patch of tall buildings in South Denver called "The Denver Tech Center". I go there every week to the Arapahoe at Village Center light rail station, but I had not checked out the surrounding area. 

The big draw is Fiddler's Green Amphitheater, a major venue for stage shows in the area. But it isn't much to look at unless you're going in and that requires a ticket.

On the other hand, there is much more packed into the area. The other day, I noticed a bronze statue of an elephant near the rail station. It was so uncharacteristic that I couldn't quite figure out what it was from a distance. It's a reminder that there's so much up the hill (probably anywhere) that you always miss things, so you have to keep going back.

Adjacent to the amphitheater is Marjorie Park with it's collection of fanciful statues, including many from the Alice stories of Lewis Carroll. There was construction going on when I visited so I could only get a few shots through the gates.

The park was originally named Samson Park after the pet Yorkie of the Museum of Outdoor Arts founders John and Marjorie Madden. The name was changed in 2015. I'll have to return after they finish construction.

Luckily, nearby Tuscany Plaza was accessible with it's collection of statues.

There is outdoor art all over the place in the Village Center/Fiddler's Green area. On the way back from the train station, I got a picture of the bear outside Plaza Tower One, the prominent 22 story building at Village Center.

In fact, you could wander around the entire Tech Center for days looking at art and architecture. 

And, of course, the views of the Rockies in this area are stunning. It's easy to get so used to them here that you forget how breathtaking they are.

The little park running through Walnut Hills is a corridor for wildlife. I spotted this big redtail hawk a couple of days ago. If you have problems seeing it, it's because they are well camouflaged in their leafy surroundings.

Speaking of wildlife, last week, a friend and I hiked up Waterton Canyon to Strontia Dam. Groups of male bighorn sheep were lounging around. I guess the females are lambing. There were plenty of birds and we saw a herd of deer on the other side of the river. 

Not a great shot but incentive to go back sometimes with my telephoto lens. There is a big difference between optical and electronic enlargement.

The Corona virus is interesting from a sociological standpoint. It's a repeat of so many epidemics that I've been through over 66 years of life but this time it's close to mass hysteria. The electronic media has opened up the world to many exciting possibilities, but it has also become a tool for manipulating huge numbers of people who have adapted quite well to it and are very open to manipulation.

It is keeping me inside more to avoid all the mad dashing around and store emptying.

Stay safe and be a neighbor for those that are more vulnerable and we'll get through it all.

Terminus: Florida Station

Florida Station is the terminus of the H light rail line that runs from downtown Denver to East Florida Avenue. I confess that I cheated and took  the R line which also stops at Florida Station. The R line passes through Arapahoe Station, which is a mile and a half from my house.

I wouldn't call Florida Station a tourist destination but it is certainly a medical destination and might be a prime birding site in the warmer parts of the year. I noted that the R and H lines also skirt Cherry Creek State Park. I'll be returning sometime.

Like all bus and train stations in the Denver area, this one has been touched by an artist. The structures are modern and the paintings are mottled transparencies on the shelters and pedestrian bridge across the highway. 

Koryn Rolstad designed both the murals, "Our Community Dance", a picture of people doing people things, and the sound in the bridge.

The bridge is a single span covered arch suspension bridge, typical for local stations.

Here's a recording from inside the bridge.

http://www.theriantimeline.com/Florida%20Noise.mp3

The noise from the highway below makes me wonder why. I guess when the traffic dies down...does that traffic ever die down?

Across the bridge is the Aurora Medical Center. It seems to be pretty highly rated with stroke and cardiac emergency centers.

A short walk west brought me to the Jewell Wetlands Park. Like many of the parks in the Denver area, this one served multiple purposes. It seems to be fairly popular with bicyclists, joggers, and groups that just want to get out. That's the people I saw there, but there are displays that are informative about the environment and a butterfly garden. It looks like the people who maintain the park (that would be the City of Aurora), intentionally create a positive environment for wildlife.

The wetlands would have been wet and more attractive,vand there would have been more wildlife if the park was not frozen over at the time. Nature has its cycles.

Here are some pictures.

The park is accessible to people who have problems getting around. The boardwalk shown above is short, but it would be a nice place to park the body and watch birds.

The rest of my day was a search for a department store my phone said was nearby - coulda fooled me, and then a trip back to my regular Walmart for monthly supplies.

Only two more terminal hikes and I'm back to the station-to-station hikes. This year I'll be taking on the RTD W line from Golden to Aurora West Station.

Don't avoid parks that are small and not-particularly-scenic. If there's one thing I've picked up in my wanderings its that every place has its points.

Earth in space 2

So, we know how long it takes Earth to rotate, that it's tilted in respect to the ecliptic, and the time it takes to travel around the sun. So how far is it from the sun?

Uhrmph...uh. Anybody got a meter stick?

Well, with some high powered telescopes and a lot of patience, we could measure the apparent distance between two stars and half a year later, on the opposite side of the sun, measure the same distance and see how much it's changed. 

Stars don't move in the sky in respect to us - not to any measurable extent, that's why they're called "fixed stars" - so any apparent motion would be due to what's called "parallax error". If you want to see parallax in action, get close to the wall on one side of an analog wall clock (the kind with hands) and read the time, then go to the other side against the wall. You'll get two different readings because you're viewing the hands of the clock from two different angles.

You can use the parallax method if you can measure very tiny angles, but the whole idea of this blog is to get by with little expense and portable equipment.

What to do?

Let's use math!

But first, the background. Back in the early 16th century, Nicholas Copernicus realized that astronomy would make much more sense if the sun were at the center of things instead of the Earth. Tycho Brahe didn't hold with Mr. Copernicus' new fangled ideas but he did have an eye for precision and cranked out a massive library of observations of the visible planets.

Johannes Kepler (born 1571), the hero of our story, did hold with Copernicus' new fangled ideas and needed his teacher, Tycho Brahe's data to prove it, but Brahe wouldn't give...until, on his deathbed, he bequeathed his library to Kepler. 

Kepler started working tirelessly on the data. It was hard for him to keep a job, and he moved from place to place. What he wanted was very pointedly something the powerful church of the time did not want, a heliocentric universe. After his benefactor, King Rudolph of Bohemia, abdicated his throne to his brother Matthias, things began to fall apart for the Kepler family, but Johannes continued working.

His way of understanding Earth in space was to, first, work out the motion of Mars from Brahe's observations. With his results, he applied them to the other visible planets to see if they worked the same way. When it was obvious that all the visible planets complied with his three laws (we talked about those in "Orbits"), he applied them to Earth. The third law is telling. It tells, in fact, how far the Earth orbits from the sun (at least, at its furthest point, and since the Earth's orbit is almost (not quite, but almost) circular, we'll go with that).

Kepler's third law says that the square of the orbital period (the time it takes a planet to orbit the sun) is proportional to the cube of the distance we want. The proportionality constant, it turns out, is equal to the sum of mass between the two bodies (the sun and the planet) multiplied by Newton's universal gravitational constant divided by 4 times pi squared (we had to wait for Newton to figure out the details). All together, the formula, 200 years in the making, is…

r is the distance we're looking for. T is the time it takes for us to travel around the sun, 365.25 days or 31,557,600 seconds. I convert to seconds because the almost-constant a, in SI units is 2.97x10-19 s2/m2. That's a rather messy looking value but remember that constants like that are mainly there to make your answer come out in the right units. I said that it's "almost constant" because it's actually a calculated value that includes the sum of the sun's mass and the mass of the object orbiting the sun. But objects in the solar system are so much smaller than the sun that their mass is negligible. The point is that the ratio of a planet's period of orbit squared to the maximum distance from the sun cubed is almost exactly the same for all the visible planets.

When I do the math, it puts us 149,675,423,264 meters or 93,003,996 miles from the sun at our maximum distance. The Wikipedia says an average of about 92 million miles and, since the Earth's elliptical orbit is almost circular, that's pretty close.

It's sorta exciting when calculations like this come out right. I used my statistical spreadsheet, DANSYS, to do the calculations but you could have done it with any spreadsheet, or even a calculator.

So, I'll see you next time and write about Florida (Station, that is) when we've moved a little further around the sun