Some prairie flameleaf sumac leaves get redder than others

Like the previous photograph, this one is from an undeveloped property off Seton Center Parkway in northwest Austin. Unlike that picture, however, this one shows redder leaves, is from November 13, and was taken with an iPhone 5s. I wasn’t out photographing at the time but had merely stopped by on my way home from something else to check how the prairie flameleaf sumac, Rhus lanceolata, was coming along, and that’s why I didn’t have my usual heavy-duty (and just plain heavy) camera equipment with me. I’d say the phone did a commendable job, wouldn’t you? Notice how the fruit clusters darken as they age and dry out.

This is the fourth and penultimate* episode in a miniseries that is carrying prairie flameleaf sumac from the beginning of August through the latter part of November.

————-

* Penultimate means ‘next to the last.’ Some people have misunderstood the word and think it means ‘ultimate, utmost, greatest, best.’

© 2014 Steven Schwartzman

What happens to prairie flameleaf sumac’s leaves in the fall

On November 17th I photographed some prairie flameleaf sumacs (Rhus lanceolata) beginning to do their autumn thing on an undeveloped property off Seton Center Parkway. As you look at what was the last of just over a hundred photographs I took on my visit to the site in northwest Austin on that sunny fall afternoon, let your eyes wander over the dense compound leaves that still favored yellow and orange over the red that is usually the destiny of this species, prime and reliable source of fall color that it is in central Texas, and prime as well in my life as a photo follower of the seasons here.

Today’s is the third episode in a little series that is carrying prairie flameleaf sumac from the beginning of August through the latter part of November. At the same time, this post begins a tribute to fall in central Texas that will go on for the next couple of weeks. Fasten your seat belts ’cause it’s going to be a colorful ride.

(I’ve decided to postpone more pictures from the Great American Southwest Adventure so you can see some of the wonderful things that have been going on in central Texas before they get too out of sync with the dates of my posts. The trip pictures are already a couple of months old, so they’ll keep just fine and should provide a good contrast with the bleaker weather we’re liable to get here in late December and January.)

© 2014 Steven Schwartzman

Prairie flameleaf sumac fruit ripening

In the last post you heard that plants in the sumac family typically make up for the small size of their flowers by producing dense clusters of them, and you saw that that’s the case with prairie flameleaf sumac, Rhus lanceolata. After fertilization, those dense clusters of flowers give way to dense clusters of small fruits, as you can verify here in a picture from September 3rd off Seton Center Parkway in northwest Austin. People have concocted sumac-ade from the fruits: I’ve had some, and I can tell you it was pretty tasty. As with lemonade, it takes a good amount of sweetener to offset the tartness of the fruit.

This is the second episode in a miniseries that is carrying prairie flameleaf sumac from the beginning of August through the latter part of November.

Prairie flameleaf sumac buds and flowers

Plants in the sumac family typically make up for the small size of their flowers by producing dense clusters of them, and that’s the case with prairie flameleaf sumac, Rhus lanceolata. This small tree is known for the rich colors its leaves turn near the end of the year, but its buds and flowers usually get overlooked; now you get to look them over in this August 1st photograph from northwest Austin. Notice the characteristic curving of the compound leaves as they arc around the buds and flowers on three sides.

This is the first post in a miniseries that will carry prairie flameleaf sumac from summer through fall.

Furry bee on cowpen daisy

Even after three-and-a-half years I still occasionally show you species that haven’t appeared here before. That’s true of this furry bee, which I don’t know how to identify, as well as the flower head it’s on, which is a cowpen daisy, Verbesina encelioides. If the genus sounds familiar, it’s because frostweed, which you’ve recently seen here with a hover fly on it and doing its ice trick, is Verbesina virginica. To my untrained eyes the two species don’t look much alike, but I’ll have to defer to botanists on that.

I took this picture on October 28th in Guerrero Colorado River Park, a location that also makes its debut here today.

© 2014 Steven Schwartzman

PhotoMath: review of a free smartphone app

As some of you might know, I taught mathematics for decades and am the author of The Words of Mathematics, a dictionary that the Mathematical Association of America has kept in print since that organization published it in 1994. Because of my math background, a friend of mine recently told me about a free smartphone app called PhotoMath and suggested that I write a review of it. As Portraits of Wildflowers is my main presence on the Internet, I’m posting the review here, with the understanding that readers interested in this blog’s normal subjects of nature photography and native plants may be surprised by the topic. On the other hand, it may be just what you’ve always wanted.

According to MicroBlink, the Croatian maker of PhotoMath, the app “uses a mobile phone camera to recognize mathematical expressions. It instantly solves a recognized expression, and displays step-by-step solution.” The program can currently handle arithmetical expressions that include fractions, decimals, powers and roots, and it can solve first-degree equations. It also recognizes some trigonometric, logarithmic, and exponential functions.

The PhotoMath app icon

Once launched, the program brings up your phone’s camera screen and imposes a conspicuous red-cornered frame in the center of it. You maneuver your phone over the page of a book or worksheet (sorry, no handwritten expressions) and use the frame to isolate a problem. You can swipe horizontally or vertically with a finger to change the dimensions of the frame to make it better fit over the shape of the printed problem. For a long expression or equation, holding your phone in landscape orientation may be better than keeping it in portrait orientation. Most problems in schoolbooks and worksheets are numbered, so you have to be careful not to include the problem number in the frame. As soon as PhotoMath recognizes a framed problem, it emits a loud click and displays the answer in a small red cartouche centered at the base of the red frame. The answer can appear so quickly it seems like magic, and that’s certainly part of the program’s appeal.

But wait, as cheesy television commercials are fond of saying, that’s not all: the lower portion of  the screen displays a larger red cartouche inviting you to press to see the steps leading to the answer.

For the expression shown here, PhotoMath takes three steps to simplify the fraction:

To maintain continuity between steps, each new screen begins with what came last on the previous screen.

Notice that as you advance from step to step the number of black dots at the lower left increases to show you which step you’ve reached, while the number of black dots at the lower right decreases with the number of steps remaining. The dots reinforce the information more largely conveyed by “step 1/3,” “step 2/3,” and step “3/3.”

The page for the last step in the working of a problem introduces a dotted line, below which the result appears. This is akin to your math teacher telling you to circle or box your answer at the end of a problem.

Speaking of students and teachers, it struck me that the latter might worry about the former using PhotoMath to do homework assignments. Here’s what the website says about that: “Let’s be honest: many kids cheat anyway, and an app which solves math problems automatically won’t make this problem worse. However, PhotoMath can be really helpful to many children when they are stuck with their homework and there is no one around to help them to figure it out. If we can eliminate kids’ frustration at the point when they can’t do anything else but helplessly stare at the book, we’ll feel awesome. It’s as simple as that.” Well, perhaps not quite that simple, but you can decide for yourself.

PhotoMath did well on problems that aren’t unusual in some way. For example, with the equation

3x + 2 = 5x – 8

it gave these steps:

3x – 5x = -8 – 2

3x – 5x = -10

-2x = -10

x = 5

So far so good, but I wondered if the app would “break” when confronted with special cases. For instance, with

4x – 2 = 4x – 2

it correctly converted the equation to

0 = 0

but left the user to interpret that truism to mean that any value of x will solve the equation. In the case of the equation

4x – 2 = 4x – 1

the app just sat there and did nothing. Not all users will understand that the lack of activity came from the fact that the equation is a contradiction (how can something be 1 more than itself?)  and therefore has no solution.

The program also did nothing when confronted with 9/0, which is undefined because no real number times the 0 in the denominator would make the 9 that’s in the numerator. The app likewise had no response to 0/0, which is undefined because any number times the 0 in the denominator would make the 0 that’s in the numerator.

For the algebraic expressionPhotoMath gave

which is true but not particularly helpful. On the other hand, when I tried

the program multiplied out the factors and correctly gave

The PhotoMath website says the app handles basic trigonometric functions, so I tried cos (30°) and was baffled by a result of .540302 rather than the correct value of approximately .866. When I tried cos 30° without parentheses the program returned an “answer” of cos 1. Then I realized that the app had treated the degree symbol as the exponent zero: 30 to the power 0 is 1, and sure enough, the cosine of 1 radian is .540302. Apparently PhotoMath evaluates trigonometric functions only for arguments that are expressed in radians.

Switching to logarithms, I was pleased to see Photomath correctly give log(2) as .30103 and ln(2) as .693147. When I thought about inverse functions of logarithms, though, and tried a natural exponential expression, the program stared at 

and did nothing. After I rewrote that as exp(3) the app correctly gave me 20.085537. Curiously, when I looked at the one and only step the program had taken to get that result, here’s what I saw:

So PhotoMath can display e cubed but can’t recognize it via the camera. Strange. It’s also quite a limitation, because math textbooks almost always use an exponential form like e cubed rather than exp(3), which is more at home in the world of computer programming.

Just as important in mathematics as e is π, but PhotoMath apparently doesn’t recognize that special constant either, because when I aimed the camera at the expression π over 2, the app interpreted it as 71 over 2 and therefore mistakenly returned a value of 35 and a half.

The PhotoMath website says the app does roots, but when I tried the cube root of 7 the program misread it as 3 times the square root of 7. When I tried the cube root of 1.331, the program misinterpreted the expression in the same way and gave an incorrect value of 3.461069; in one instance (I tried this expression several times), it even threw away the decimal point in 1.331 and came up with a false result ten times as large.

PhotoMath’s success when I stuck to arithmetic expressions was pretty good. The compound fraction given as a sample on the website,

offered no trouble when I tried it.  The app also did a good job with first-degree equations, even a disguised one like

for which it returned the correct value of x = nine fourths. Systems of linear equations aren’t supported, however.

When PhotoMath accesses your phone’s camera, in addition to the red frame at the center of the screen it shows four icons across the bottom, which you can see in this view from the program’s help section:

The History button takes you to a list of recently read problems. I don’t know how many items the list can retain, but after trying out the program for a day I found that my list had more than 50 items in it. Tapping on an item brings back all the solving steps, so you can review them later. The Steps button brings up the steps in a problem that has just been solved.

I assume the Light button is intended to turn on the phone’s light in case the page you’re aiming the camera at isn’t bright enough, but I never could get the light to come on with my iPhone 5 running the latest version of iOS 8. The Help button offers some very basic information about using the program.

Given PhotoMath’s hit-and-miss record when I put it through its paces, I’m tempted to say caveat emptor, let the buyer beware, but this is a free app, so no money is at stake. Still, caution is in order, and users should examine results for plausibility: the cube root of 1.331 couldn’t possibly be the 3.46 that PhotoMath claimed, or any number more than a bit larger than 1 (in fact the cube root of 1.331 is exactly 1.1). A good strategy might be to look at all the steps PhotoMath offers as its justifications for an answer, because then any misinterpretation is likely to be obvious (like π being misread as 71).

On October 23, 2014, the blog on the PhotoMath website glowingly announced that “the PhotoMath video on Vimeo has very quickly reached 2 million views. Our web page has over 9000 page views each minute, and the iOS app alone was downloaded more than 1.6M times in less than three days, becoming the top free app in most countries around the world.” That’s pretty impressive, and I encourage you to head over to PhotoMath and increase those numbers by trying out the program for yourself.

PhotoMath is currently at version 1.1.1 and is available for Apple (it requires iOS 7.0 or later; is compatible with iPhone, iPad, and iPod touch; is optimized for iPhone 5 and iPhone 6) and for Microsoft (Windows Phone 8 or 8.1). The makers of PhotoMath say that an eagerly awaited Android version should launch in early 2015.

Asters in their own right

In the previous picture, out-of-focus asters (Symphyotrichum spp.) were fine as a background, but if you wanted to see what these dense flowers look like in their own right, here’s your chance. The green vine snaking across the top of the photograph is the appropriately named greenbrier, Smilax bona-nox.

Once again the date was October 28th, and the location Springfield Park in southeast Austin. I saw a few of these asters as recently as two days ago, even after a cold spell hit us last week (but then the afternoon high bounced back up to 81°F, or about 27°C, three days ago).

© 2014 Steven Schwartzman

Eastern gamagrass with its own flowers and others

Grasses are flowering plants—honest, even if their flowers are tiny and don’t look like roses or daffodils or daisies. Here you see a leaning stalk of eastern gamagrass, Tripsacum dactyloides (which I believe is Austin’s tallest native grass), with a goodly number of its pendulous male flowers. Beyond those foreground flowers are similarly numerous flower heads of asters in the genus Symphyotrichum.

The date was October 28th, and the location Springfield Park in southeast Austin.

© 2014 Steven Schwartzman

Ocotillo Y (why not?)

On September 29th at Lost Dutchman State Park outside Phoenix I photographed this ocotillo, Fouquieria splendens. Recent heavy rains had caused these spindly plants of the Sonoran Desert to put out a dense covering of leaves to do some photosynthesis while the photosynthesizing was good, which is to say before the usual drought of the desert set back in and the plants would shed their briefly used leaves as quickly as they’d put them on.

If you’d like a reminder of what the bright red flowers of ocotillo look like, you can check out a post from west Texas this past spring. Notice how the ocotillo in that earlier photograph didn’t have a single leaf on its dry branches, but that didn’t stop it from putting out flowers.

———–

This is another entry from the saw (as in the past tense of see) part of the see-saw that’s been bouncing back and forth between pictures from my trip to the American Southwest in late September and more-recent pictures showing what’s been going on in Austin.

© 2014 Steven Schwartzman

Three plus one equals four, but the flowers of trailing four o’clocks are three in one

On my way along the Apache Trail from Phoenix to Canyon Lake on September 29th, I couldn’t help noticing ground-hugging colonies of magenta flowers by the side of the road (see the first photo, which looks mostly downward). Thanks to George Miller of the Native Plant Society of New Mexico, I now know that these are trailing four o-clocks, Allonia incarnata. Here’s a closer look at one of the flowers.

The scientific name of this wildflower teaches us that plants in the genus Mirabilis aren’t the only ones that people have called four o’clocks (even if some wildflowers in that genus go by other popular names as well, like the angel’s trumpet you recently saw). I’ve learned that the genus Allonia includes just two species, and that what appears to be a single flower is actually three flowers; the six lobes in the upper right of the close-up photo delineate one of those flowers, and you can count six lobes clockwise and counterclockwise from that flower to see the other two  flowers that complete this triune inflorescece. That just gave me an idea for what I would call an inflorescent bulb: turn on the power and out comes a flower. Of course some bulbs already give rise to flowers, just not at the speed of light. Oh well, flights of fancy aside, you’re welcome to read a down-to-earth Wikipedia article about the two-species Allonia genus.

———–

This is another entry from the saw (as in the past tense of see) part of the see-saw that’s been oscillating between pictures from my trip to the American Southwest in late September and more-recent pictures showing what’s been going on in Austin.

© 2014 Steven Schwartzman