Possumhaw’s time to shine again

While walking along the trail that parallels the south shore of Lady Bird Lake yesterday afternoon I spied some possumhaws, Ilex decidua, with lots of little fruits on them. This iPhone picture lets you see the colorful view but you’ll have to imagine feeling the 72° (22° C) that the temperature got up to.

© 2015 Steven Schwartzman

A lush and colorful revelation

In the last post you saw poison ivy in its vine form turning colors, as the plant usually does here in late November or early December. Earlier on that same morning of November 25th I’d turned north off 45th St. onto Perry Lane when suddenly the blaze of colors shown here greeted me in the front yard of a house. All that wonderful color was coming once again from poison ivy (Toxicodendron radicans). The plant was so lush I couldn’t initially tell what form it had taken, but eventually I peered through openings between the leaves and found that the poison ivy appeared to be in its bush form. What’s more, it had taken root beneath an often-planted invasive type of non-native tree (alas!) and had almost completely eclipsed the lower part of it (yay!). Notice that not only can poison ivy’s leaflets turn yellow or yellow-orange, as you saw last time, but they can go all the way to red.

Three days later I passed back by and knocked on the door of the house. I asked the woman who answered whether she knew that all those pretty-colored leaves were poison ivy, and in fact she didn’t. She added—I think separately from the fact that the plant had turned out to be poison ivy—that she didn’t find the leaves attractive. Oh well, one person’s gorgeous fall foliage is another person’s blah.

Another six days later, on the morning of December 4th, I happened to drive past the house again just when a fire engine and an EMS vehicle were out front with their lights flashing. I saw that the woman I’d spoken with a week earlier was being wheeled out to the ambulance on a gurney. She was conscious and didn’t seem to be in distress, but obviously something must have happened to her.

This is the second recent photograph I’ve posted from an iPhone 5s. When I took pictures at this site the first time I didn’t have my regular photo equipment with me, but I was happy enough with the phone’s version that I didn’t feel any need to redo it—aside from which the leaves and the weather were both duller three days later when I stopped by the second time.

© 2014 Steven Schwartzman

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.

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

© 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.

Mortgages, Investments, Business Services, Retirement, and …

Click for greater size and clarity.

… birds. Yes, there I was on August 21 at the drive-up window of a bank in my neighborhood when I noticed a couple of birds on the hedges just ahead of me on my left, adjacent to the bank’s reflective windows. I pulled forward a little to be alongside the birds and I kept my car window up in hopes that its tinting would keep the birds from seeing me inside. Then I reached for my iPhone*, put it up against the car window, and took three pictures, of which this was the best. Whether anyone inside the bank was looking out and watching me do that, I don’t know; I couldn’t see inside the bank because its windows were too dark and reflective (and they also made my car, which is pure white, come out gray in this photograph). After this third picture I lowered the car window to see if I could get a clearer shot of the closer bird, but it got spooked and flew away, followed by the other bird. Their dark reflections went with them.

As I’ve said several times in this blog, I rarely include human elements in my nature pictures, but once in a while I find a reason to make an exception.

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* I wonder if the white spot visible in my car’s window was the iPhone 4s’s puny attempt at a flash.

© 2012 Steven Schwartzman

A wider view of dense bluebonnets

Click for greater size and clarity.

By coincidence, my wife Eve has been attending a three-day seminar at the J.J. Pickle Research Campus, the place outside of which I photographed a bluebonnet colony on March 26. When I went to pick her up there yesterday at 5 PM, I took a picture of an even larger colony of bluebonnets in a field inside the campus, so here’s a broader view than the one you saw in the last post.

This photograph has the distinction of being the first one I’ve ever shown here that I took on my iPhone 4s. I’m not about to abandon my fancy DSLR (digital single-lens reflex camera), but I think you’ll agree that the phone did a pretty good job. It even attached the latitude, longitude, and altitude of the location (30,23.27N, 97,43.67W, 243.00 m), though it didn’t record the scent of the bluebonnets. Guess we’ll have to wait for the iPhone 5 for that feature.

© 2012 Steven Schwartzman