Portraits of Wildflowers

Perspectives on Nature Photography

Posts Tagged ‘mathematics

Pi Day

with 47 comments

Some math-minded folks refer to today, 3/14, as Pi Day because 3.14 is the approximate value of π, the ratio of a circle’s circumference to its diameter. In other words, if you could lift a diameter out of a circle, bend it to match the curvature of that circle, then lay it back down onto the circle, it would take about 3.14 such curved segments to go completely around. π is what mathematicians call a transcendental number; one consequence is that we can’t express its exact value with a terminating decimal or even a repeating decimal (as, for example, 1/8 = exactly 0.125 and 1/11 = 0.09090909…).

What’s all that got to do with this opening four-nerve daisy (Tetraneuris scaposa) that I photographed in my neighborhood four days ago? Well, 4 is a number, right? And you’ve gotta admit that the sunny yellow flower head does a good job of suggesting a circle.

© 2019 Steven Schwartzman

Written by Steve Schwartzman

March 14, 2019 at 4:46 AM


with 35 comments

Here on day 11 in month 11 of the year is something appropriate for veterans of arithmetic:

Written by Steve Schwartzman

November 11, 2018 at 5:03 AM

Bug nymph on four-nerve daisy

with 36 comments

In contrast to the willful four-nerve daisy flower head (Tetraneuris linearifolia) you saw last time, the flatness of this one that I found on the same April 1st outing had me aiming straight down at it.

You’ll remember that each “petal” of a daisy is actually an independent flower known as a ray flower. The rays (14 in this case) ray-diate out from the flower head’s center, which is made up of many smaller individual flowers of a different type, known as disk flowers. It’s common in daisies for the disk flowers to form overlapping spirals, some of which go out from the center in a clockwise sense, and others in a counter-clockwise sense. If you count the number of disk-flower spirals in each direction, you typically get consecutive Fibonacci numbers. There’s a confirmation of that in the following enlargements of this four-nerve daisy’s disk. Go ahead, count the number of spirals going each way and you’ll see:

In the unlikely event that anyone ever asks you if daisies know how to count, you can confidently and Fibonaccily say yes.

© 2018 Steven Schwartzman

Written by Steve Schwartzman

April 13, 2018 at 4:35 AM

Sunday sunset 4

with 32 comments

On each of the four Sundays in January you’ve seen sunset pictures from the state whose license plates proclaim it the Land of Enchantment. Now that today’s post concludes the series, you’re welcome to look back at the other photographs that have appeared here from June 10, 2017, at Camel Rock, 11 miles north of downtown Santa Fe.

Because the first Sunday sunset picture this month appeared on January 7, and because there are 7 days in a week, all of the January pictures in this sequence came on dates divisible by 7:  7, 14, 21, 28. Speaking of divisibility, if you divide 1 by 7, and then 2 by 7, etc., to convert the fractions to decimals, you’ll find that the sevenths give the following infinitely repeating six-digit cycles:

Do you see the cyclical nature of those decimal expansions, with each one consisting of the same digits in the same order, only starting at a different place in the cycle?

But wait! The columns want some attention, too. Notice that reading down the first column of decimal digits is the same as reading up the fourth column? Likewise for the second and fifth columns, and also for the third and sixth columns.

There’s more that could be said, but for now I’ll let the sun set on these mathematical pleasures and not take you further into seventh heaven.

© 2018 Steven Schwartzman

Written by Steve Schwartzman

January 28, 2018 at 4:58 AM

Aztec dancer and ant

with 71 comments

Aztec Dancer Damselfly with Ant 9632

I took this photograph close to a waterfall off Harrogate Dr. in northwest Austin last year on 7/24. Whether the ant ran any risk of getting eaten by the Aztec dancer damselfly, Argia nahuana, I don’t know, but the date reminds me of something I do know, namely that 7 and 24 are the perpendicular sides of a 7-24-25 right triangle because 7 squared plus 24 squared equals 25 squared. Other right triangles with the shortest side an odd number are  5-12-13,  9-40-41,  11-60-61,  13-84-85,  and the familiar 3-4-5. Can you figure out how to get the two longer sides of each right triangle of this type if you know only the shortest side?

(Speaking of math, did anyone notice that the number 63 that played a role in yesterday’s post can be written in base 2 as 111111?)

© 2015 Steven Schwartzman

Written by Steve Schwartzman

July 27, 2015 at 5:32 AM

New Zealand: Fibonacci comes to the Marlborough rock daisy

with 16 comments

Marlborough Rock Daisy Receptacle and Shadow 5460

One native plant that caught my attention at Otari-Wilton’s Bush in Wellington on February 20th was the Marlborough rock daisy, Pachystegia insignis. In the central disk of these seed head remains you can confirm the presence of consecutive Fibonacci numbers: I count 13 clockwise spirals and 21 counter-clockwise spirals (I believe New Zealanders say anti-clockwise), and I’ve thrown in 1 conspicuous shadow at no extra charge.

If you’d like to confirm the Fibonacci counts for yourself, click on the disk below for an enlargement. I find it easier to pick out the counter-clockwise spirals, but both are there.

Marlborough Rock Daisy Center 5460

© 2015 Steven Schwartzman

Written by Steve Schwartzman

June 18, 2015 at 5:11 AM

PhotoMath: review of a free smartphone app

with 32 comments

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 expressionx-2yPhotoMath gave

x - 2y squared

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


the program multiplied out the factors and correctly gave

Screen Shot 2014-11-21 at 1.35.09 PM

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 e cubed

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,

Screen Shot 2014-11-21 at 2.38.23 PM

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

Screen Shot 2014-11-21 at 2.23.12 PM

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.

Written by Steve Schwartzman

November 26, 2014 at 11:17 AM

A warmer nod to arithmetic

with 35 comments

Seventeen months ago today, in a post entitled “A warm nod to arithmetic,” I wrote the following:

I’m sure you noticed something curious in the last post: the equivalent (rounded to the nearest whole degree) of Thursday’s warm high temperature of 82°F was 28°C, a number consisting of the same two digits but in reverse order. Some of you must have rushed to Twitter and Facebook to tell all your friends, who must have been thrilled to hear it.

Ever alert math teachers would interrupt their classes with a question now: “Students, are there any other two-digit pairs of equivalent Fahrenheit ~ Celsius temperatures with reversed digits like that one?” What do you think, readers?

All of you who read that post were probably too overwhelmed with emotion by the equivalent pair of 82°F and 28°C to do any further looking, and no answer to my question was ever forthcoming (or ever forthcame, for that matter). As you’ve all been waiting with bated (not baited) breath and have long since turned blue, I’d better tell you the answer. Yes, there is another reversed-digits pair of Fahrenheit ~ Celsius temperatures that are equivalent (when rounded to the nearest whole degree): 61°F and 16°C. Now go give a shout-out and spread the news to everyone you know. Tell ’em Steve sent you.

You can also tell them that Steve sent you the photograph below so you wouldn’t feel bereft of a picture today. I hope arithmetic hasn’t made your head feel the way this Texas thistle head (Cirsium texanum) looks.

Texas Thistle Flower Head Decomposing 5161

© 2014 Steven Schwartzman

Written by Steve Schwartzman

August 10, 2014 at 5:57 AM

%d bloggers like this: