Portraits of Wildflowers

Perspectives on Nature Photography

Looking back at the Sandia Mountains

with 20 comments


Late in the afternoon on October 15th we drove up to Sandia Crest, which at 10,679 ft. looms large to the northeast of Albuquerque. The previous post showed views from there. The next morning in our hotel’s parking lot I noticed that the view back toward the mountains, now covered with fog and clouds, was dramatic—at least if I could ignore light poles, buildings, billboards, highways, and other trappings of the city. To exclude as much of that as possible, for the top picture I zoomed my telephoto lens to its maximum focal length of 400mm. With a change of scale and locale you might see an ocean wave breaking near the shore. Three-quarters of an hour later and miles further north as we wended our way toward Santa Fe, the land added color to the still-shrouded mountains.








Look at this:


1    3    5    7    9    11    13    15    17    19    21    23    25    27    29  …


If I asked you what it is, you’d say it’s the odd numbers.

(Or if you wanna get fancy, you’d say it’s the positive odd numbers.)

You’ve been seeing these numbers for a long time, ever since you were in elementary school.

Now here’s something you might never have noticed.


The “sum” of the first 1 odd number is 1, which is 1 x 1, or 1 squared.

The sum of the first 2 odd numbers is 1 + 3, or 4, which is 2 x 2, or 2 squared.

The sum of the first 3 odd numbers is 1 + 3 + 5, or 9, which is 3 x 3, or 3 squared.

The sum of the first 4 odd numbers is 1 + 3 + 5 + 7, or 16, which is 4 x 4, or 4 squared.

The sum of the first 5 odd numbers is 1 + 3 + 5 + 7 + 9, or 25, which is 5 x 5, or 5 squared.

The sum of the first 6 odd numbers is 1 + 3 + 5 + 7 + 9 + 11, or 36, which is 6 x 6, or 6 squared.

The sum of the first 7 odd numbers is 1 + 3 + 5 + 7 + 9 + 11 + 13, or 49, which is 7 x 7, or 7 squared.


Now if I skipped ahead 93 lines and asked you what the sum of the first 100 odd numbers is, you wouldn’t have to do any adding at all. You’d chime right in and say with verve and élan that the sum of the first 100 odd numbers is 100 x 100, or 10,000.*


If you’d like a nifty visual explanation for why the sum of
the first however many odd numbers is always a square, here it is:



But if you’d also like an explanation of where the word nifty came from, you’re outta luck: no one knows.


* One day in the early 1970s I was in a supermarket in my home town of Franklin Square, New York. As I walked down an aisle in that large store I passed a woman who was telling her young daughter that ten times ten is a hundred and a hundred times a hundred is a thousand. Why she was saying that, I don’t know; why I didn’t intervene and correct her I also don’t know. Evidence points more to politeness than a lack of boldness. That’s because in the same supermarket minutes later when I heard two women talking about pressure cookers, with each saying they had one but didn’t use it, I approached them and asked if I could have those pressure cookers that they didn’t use. I ended up several days later with two pressure cookers, one of which, a fancy stainless steel model with a copper bottom, had been a wedding present and was essentially brand new. I used it for decades until finally part of the handle came loose and the rubber gasket no longer sealed properly, and replacement parts were no longer available for such an old pressure cooker.


© 2022 Steven Schwartzman





Written by Steve Schwartzman

November 23, 2022 at 4:26 AM

20 Responses

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  1. It does look like Surf’s Up!

    Robert Parker

    November 23, 2022 at 8:12 AM

    • Who’d’ve thought of Albuquerque as Surf City?

      Steve Schwartzman

      November 23, 2022 at 9:03 AM

      • Yeah a pretty dry place. When I was there as a kid, visiting relatives in the area, it was over over 100F every day and getting up in the relatively cool Sandias was a big treat.

        Robert Parker

        November 23, 2022 at 9:21 AM

        • In English we keep as cool as a cucumber. In Albuquerque you kept as cool as a sandía (watermelon) in the Sandía Mountains.

          Steve Schwartzman

          November 23, 2022 at 10:07 AM

  2. The fog on the mountains is neat.
    Math and numbers baffle me most the time, but I do appreciate how like magic and perfect they are!


    November 23, 2022 at 8:20 AM

    • We don’t get much fog near where I live, so I appreciate it whenever I find it.

      Your use of “perfect” inadvertently touches on math in another way. So-called perfect numbers are those like 28 in which all the smaller (positive) divisors add up to the number itself. The smaller numbers that go into 28 are 1, 2, 4, 7, and 14; add those up and you get 28.

      Steve Schwartzman

      November 23, 2022 at 9:07 AM

  3. Zooming in on the clouds was a clever way to avoid the ugly billboards or the telephone poles. Thanks for the math lesson. Steve!

    Peter Klopp

    November 23, 2022 at 8:53 AM

    • Even where I couldn’t fully exclude junk from my pictures, I composed in such a way that I could later crop off the stuff I didn’t want. With 50-megapixel images, you still have plenty left after cropping even half of the pixels off.

      Steve Schwartzman

      November 23, 2022 at 9:11 AM

    • I just checked and found that in the top picture I was able to shoot over all the people-related objects and didn’t need to crop at all.

      Steve Schwartzman

      November 23, 2022 at 12:01 PM

  4. Uau. The clouds cascading down the mountains is something I saw once back in Itaipava during a hike. Beautiful phenomenon.

    Alessandra Chaves

    November 23, 2022 at 11:45 AM

    • It is a beautiful phenomenon. I’m happy I could get good pictures from the top of the mountain late one afternoon and again from its base the next morning.

      Steve Schwartzman

      November 23, 2022 at 12:03 PM

  5. The two women in the supermarket must have been glad to be relieved of the pressure of their pressure cooker problems. Do you still use a pressure cooker? My flatmate and I used a pressure cooker in the 70s. The photos are lovely. It’s hard to believe there were poles and billboards etc nearby.


    November 24, 2022 at 1:33 AM

    • I was only too happy to take the pressure off them and bring it into my kitchen. Yes, we still use a pressure cooker. Maybe seven years ago we bought a stainless steel set with a single lid that fits each of two stovetop pots, one full height and the other half-height for smaller quantities. A few years after that a company called Instant Pot came out with electric pressure cookers that keep all the steam inside and don’t have the sh-sh-sh of slowly released steam from the older kind. We bought one of those and now primarily use it.

      As for the top picture, you’d never suspect I was in a hotel parking lot surrounded by cars and within sight of poles, wires, buildings, etc. The other picture, though taken by the side of a busy highway, was easier, because we’d gotten out of Albuquerque and I was aiming away from the traffic.

      Steve Schwartzman

      November 24, 2022 at 1:58 AM

  6. That is “sum” great fog hanging over the mountains and must have been enjoyable to watch as it swelled across them like waves.

    Steve Gingold

    November 24, 2022 at 6:55 AM

    • Just as there’s carrying in addition, I like your carrying over of “sum” into a comment on nature. The fog didn’t change much on a scale of minutes but as we traveled north I scrambled to find unencumbered spots from which to get new shots of the mountains.

      Steve Schwartzman

      November 24, 2022 at 9:24 AM

  7. These photos are from territory I have visited, and enjoyed immensely. You probably visited Chimayo on one of your trips. One of my favorite memories is taking the high road to Taos: names like Truchas and Trampas still evoke some of the sights and people I encountered. That’s an area I’d enjoy visiting again — maybe next fall.

    The layers of the second photo are remarkable not only for their color but also for their texture. The layer just above the golden land looks like the mouton coat hanging in my closet. Despite years in Texas and little possibility I’ll ever wear it again (thanks to my aversion to traveling north in winter), it’s so beautiful I can’t get rid of it.


    November 24, 2022 at 10:32 AM

    • You’re correctly anticipating. Posts next week will feature The High Road to Taos Scenic Byway and Nambé and Chimayó, even if on this trip we didn’t quite make it to Taos.

      I’m not sure I’ve heard of a coat described as mouton. It reminds me of a French expression, “Retournons à nos moutons,” literally “Let’s return to our sheep,” but conveying the meaning “Let’s get back to the subject.” I don’t blame you for not wanting to travel north in the winter, but our experience on this trip confirmed that in mid-October in the high altitudes of northern New Mexico you could still easily have occasion for a good coat before the day itself warms up. It’d be a great balance between warm and cold.

      Steve Schwartzman

      November 24, 2022 at 10:48 AM

      • And that’s exactly what a mouton coat is: sheepskin sheared closely enough that it resembles seal, or another fine fur.


        November 24, 2022 at 10:49 AM

        • You’re probably aware that after the Norman invasion English ended up keeping native names for animals like sheep and cow and pig and calf while adopting versions of the French equivalents—mutton, beef, pork, veal—to designate the meat of those respective animals.

          Steve Schwartzman

          November 24, 2022 at 11:12 AM

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