Portraits of Wildflowers

Perspectives on Nature Photography

Texas flax

with 45 comments

From April 9th along FM 1431 north of Marble Falls comes a colony of what I take to be Linum hudsonioides, known as Hudson flax or Texas flax, that turned the land yellow. Below is a closer look that includes some flowering and budding globes of antelope horns milkweed, Asclepias asperula.

The third picture offers an even closer view so you get a better sense of what these flax flowers are like. The yellow flowers without red centers are a kind of bladderpod (Physaria sp., formerly Lesquerella).

A theme I’ve been pursuing for over a week now is that it’s common to hear politicians and activists bandy about the phrase “common sense,” which I find to be a loaded and misleading term because some or even many things that a majority of people believe to be common sense can be shown not to be true.

Here’s an example from geography. Suppose you have access to a list of all the rivers (including streams, creeks, etc.), in the United States, along with the length of each one (rounded to the nearest whole mile). For example, that list would include:

The Missouri River (various states): 2341 miles.
The Rio Grande (various states): 1759 miles.
The Ohio River (various states): 979 miles.
The Yellowstone River (mostly Wyoming): 678 miles.
The Cache River (Missouri): 213 miles.
The San Marcos River (Texas): 75 miles.
The Scott River (California): 60 miles.
The East Mancos River (Colorado): 12 miles
The Chelan River (Washington): 4 miles.
The Kisco River (New York): 3 miles.

There’d be thousands of rivers in the full list. The number for the length of each river has a first—and in some cases only—digit. Now here’s the question: of all those thousands of lengths, what portion (or fraction or percent) of them have 1 as their first digit? “Common sense” would lead many people to think as follows: “Rivers are natural phenomena, free from any human bias. They come in all sorts of lengths, from very short to very long, so it seems the length of a river is as likely to begin with any digit as with any other. There are 9 possible first digits (0 can’t be a first digit for a length), so on average 1/9 of the lengths, or about 11%, would have 1 as their first digit. The same would be true for each of the other possible first digits.”

Alas, rivers don’t have that sort of “common sense.” Dumb aqueous brutes that they are, they keep on going with the flow in their own stubborn way. If you could see the list of all the river lengths, you’d find that about 30% of them begin with a 1, nearly 18% with a 2, and so on down the line in decreasing fashion, with not even 5% of the lengths beginning with a 9.

Ah, you say, maybe that’s because Americans are recalcitrant and cling to antiquated measures of length like inches, feet, yards, and miles. Surely there’d be “equity” (oh, that horrid word, which means forced sameness of outcomes for groups) if we did our measuring in civilized kilometers rather than hillbilly miles. It turns out that if you converted miles to kilometers, most individual river lengths would end up having a different first digit than before, yet amazingly the first digits as a group would still follow the same distribution, from 1 as the most common down to 9 as the least common!

This phenomenon, which holds for many things other than lengths of rivers, has come to be known as Benford’s Law. You’re welcome to read more about it. (And we should add that Benford’s Law follows Stigler’s Law, which “holds that scientific laws and discoveries are never given the names of their actual discover.”)

© 2021 Steven Schwartzman

Written by Steve Schwartzman

April 16, 2021 at 4:36 AM

45 Responses

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  1. It does my heart good to see wildflowers dominating the landscape like this!

    Ann Mackay

    April 16, 2021 at 5:01 AM

    • Me too. The land off to our left as we drove looked so yellow that I felt compelled to turn around and go back. The property was fenced, and I had to lean over barbed wire for all my pictures. Several times I had to work my shirt loose from a barb afterwards. It’s too bad I couldn’t wander freely to get closer and compose the way I wanted do.

      Steve Schwartzman

      April 16, 2021 at 7:33 AM

    • Me too Ann x


      April 16, 2021 at 6:08 PM

  2. A very pretty sight, it’s a pleasant, sunshiny sort of yellow.
    Here’s a “common sense” question that occurred to me when reading your post. How long is Six Mile Creek (Ithaca, NY)? Or Ninemile Creek (Syracuse, NY)? Or Ten Mile River (Dutchess Co. NY)? The answers do not, of course = the name.
    I was also curious about where people are cultivating flax Linum usitatissimum, for linen and flaxseed, apparently some farms in Texas do grow it, but mostly it’s northern states, N. & S. Dakota, Montana, Minnesota.

    Robert Parker

    April 16, 2021 at 6:02 AM

    • Names aren’t always accurate, are they? Think about some very tall people whose friends humorously call them Shorty. I wonder if the creeks you mentioned got their names from being a certain number of miles away from some starting point. When I visited Detroit for the first time I remember passing through the northern reaches and noticing east-west streets named X Mile Rd, where X was the number of miles north of the city (someplace downtown, I presume).

      As for the most useful (usitatissimum) flax, I’m afraid I can spin no stories for you. You’ve found out more about it than I knew.

      Steve Schwartzman

      April 16, 2021 at 7:44 AM

    • When I first met our native flax, I learned about Linum usitatissimum. Its flowers are blue, and it’s native to Europe. As far as I know, all of our native species (or at least the three that I know) are yellow with slight variations in leaves, center color, and so on. The USDA map shows the introduced species in Texas, but it’s only in a dozen or so counties. It used to be common in New England, thanks to the colonists, but it’s slowly disappearing there.


      April 16, 2021 at 11:54 AM

      • One of my grandmothers was a lifelong gardener – flowers, vegetables, roses – and she always had some perennial flax growing, I don’t know what type, in an area with only blue flowers. If you’ve ever read about the process for making linen, in the old days, it was crazy labor-intensive. Thank heavens for Whitney’s cotton gin.

        Robert Parker

        April 16, 2021 at 12:05 PM

  3. If there were infinitely many rivers, common sense would tell me that there would be a more even distribution of the digits. But the number of rivers is rather finite. So from a statistical point of view, the results appear to be skewed. Please let me know if you find fault with my reasoning, Steve.

    Peter Klopp

    April 16, 2021 at 8:34 AM

    • On the contrary: the greater the number of pieces of data you had for phenomena that obey this rule, the closer they would come to following the exact mathematical distribution, with 1 as by far the most common first digit, down to 9 as the least common first digit. You can read up on Benford’s Law to see why that is so.

      In contrast, if you were looking at last digits, you’d be correct, and the 10 possible last digits would be equally distributed.

      Steve Schwartzman

      April 16, 2021 at 8:45 AM

      • Interesting! Thanks for the elucidation, Steve!

        Peter Klopp

        April 16, 2021 at 8:48 AM

        • Sure thing. Benford’s Law is not an instance of “The first shall be last and the last shall be first.” The two sets of digits behave differently. That’s the surprising thing.

          Steve Schwartzman

          April 16, 2021 at 8:56 AM

  4. So beautiful the landscape covered in gold. Strange about the rivers…

    Alessandra Chaves

    April 16, 2021 at 8:39 AM

    • Yes, hooray for golden landscapes. And Benford’s Law applies to many things other than river lengths. It’s a fascinating topic to read up on.

      Steve Schwartzman

      April 16, 2021 at 8:47 AM

  5. Gorgeous wildflower carpet, Steve, I enjoyed you bringing us in gradually.

    Jet Eliot

    April 16, 2021 at 8:58 AM

    • It’s good to hear you find the gradual zooming in effective. If it hadn’t been for the barbed wire, I’d have been able to get in close for some individual portraits as well.

      Steve Schwartzman

      April 16, 2021 at 9:00 AM

  6. I’ve never seen such a spread of flax, and I wondered if this species might colonize more readily than the species I’m most familiar with: Linum alatum (Rockport and Palacios) and L. berlandieri (Brazoria, Kerrville and surrounding area). That red center sure is bright; it surely must be one of the field marks that helps to distinguish this one from other yellow flaxes.

    While driving last week, I thought I’d spotted a group of antelope horns, but I didn’t turn around to check. Seeing your photo, I suspect that’s exactly what I saw. I’ve not seen any others yet this season, but they must be around.


    April 16, 2021 at 12:03 PM

    • I’ve been seeing antelope horns here and there around Austin for weeks. On this April 9th trip further west we saw it in a bunch of places. Antelope horns tends to form “fairy rings” like the one in the second picture. As for the flax, I’m not used to seeing a broad swath of it like this, which of course is what made me stop. As you said, all the native species I’ve seen in Texas have yellow flowers with some sort of reddish markings in the center.

      Steve Schwartzman

      April 16, 2021 at 1:23 PM

  7. That’s quite a field full of yellow.

    Steve Gingold

    April 16, 2021 at 5:33 PM

  8. This reminds me of our super bloom here in California a couple of years ago. An amazing carpet.

    Michael Scandling

    April 16, 2021 at 6:27 PM

    • I’m happy with the glorious wildflowers in Texas—which doesn’t stop me from hoping to someday see your California superbloom.

      Steve Schwartzman

      April 16, 2021 at 6:30 PM

  9. wonderful creatures….
    love the pictures

    kamis mirin

    April 16, 2021 at 11:09 PM

  10. I have never seen a field of flax like that! It makes for such a lovely sea of yellow. We see a few plants here and there on our property, and I always mow around them, leaving them to go to seed. I’ve done that with wildflowers of all sorts over the years. This year I have huge areas of yarrow. Maybe someday it will cover the pasture!

    I can’t say I’ve ever seen antelope horns… they’re beautiful too!


    April 17, 2021 at 8:24 AM

    • Antelope horns is by far the most common milkweed species in Austin and surrounding areas. I remember finding some in the late 1970s or early 80s, long before I got interested in native plants, and being attracted to the strangeness. You can get a closer look at one of those flower globes in a post from 2014:

      Antelope-horns flower globe

      If you drag the slide up to zoom in on the map at https://plants.usda.gov/core/profile?symbol=ASAS you’ll see that it has been found in quite a few Oklahoma counties.

      As for the flax, I’d previously been like you, having seen only isolated flax flowers or small groups. Maybe this species is more given to forming large colonies than other flax species.

      Steve Schwartzman

      April 17, 2021 at 9:07 AM

  11. Never heard of Benford’s Law, or the concept, and found it very interesting. I read the article at the link and am still none the wiser as to why it should be though!

    • One way to think of it is that there are many more short rivers than long ones. The longer a length you want to find an example of, the fewer there will be to choose from. You’ll find plenty of rivers 10, 11, 12, 13, 14,… 19 miles long before you even get to any whose length begins with a 2.

      Steve Schwartzman

      April 17, 2021 at 1:04 PM

  12. I’ve bookmarked the two sites you mention for a later read. The toad flax is a beautiful flower. For some reason I thought it was blue, but stand corrected here.

    Lavinia Ross

    April 18, 2021 at 11:42 AM

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