The golden ratio, which can be found in flowers like the aster goldenrod, is understood to be a fundamental pattern of beauty in nature and art. Photo by Dana Wilde

“Billions and billions of stars,” Carl Sagan famously intoned decades ago, meaning: The universe is immense. And that was just in outer space.

Down on Earth there are stars by the myriadic welter too, all summer through to fall. The biggest, brightest of them are the sunflowers, leaning east to west from morning to night and soaking up sunlight with big yellow-gold wheels. Their predecessors are hawkweed, dandelions and ox-eye daisies — all in the family Asteraceae, from the Latin astrum, Greek astro, meaning “star.” Black-eyed Susans, powder-blue chicory, tansy buttons, fleabanes, and by late September violet-blue New York asters, deep blue-violet New England asters.

And then there’s the goldenrod.

More than 100 different kinds, with cascades of flowers close-packed on stems that seem to be overflowing rather than sunning. Goldenrods are asters too — the cascades are made of tiny yellow blossoms that themselves are stars. Rough-stemmed goldenrod grows in every uncut field from July to October, and also seaside goldenrod, blue-stemmed, stiff, downy, tall.

Long ago goldenrod got badmouthed for causing hay fever, but it doesn’t — ragweed’s the culprit. Its green flowers spill pollen at the same time as goldenrod blossoms. In fact, by age-old tradition goldenrod leaves make a tea you can drink for a sore throat. The better to sing the Solidago electric.

Every autumn, in the gold September light, they seem more and more gorgeous. At a certain point they become intoxicating. They seem to march in ragged clusters up and down hillsides, bright yellow in the sun and ageless. Whole sections of fields get flooded with gold, like sprays from the Milky Way washing through late-summer constellations. It’s hard not to think of them as small galaxies growing in dry grass and sunlight on the edge of the woods.

Around 1600, Johannes Kepler, while searching for the harmonies in the celestial spheres long before anyone knew what a galaxy was, noticed that flower parts grow in spirals. Botanists call it spiral phyllotaxis: Leaves on a stem and the structures inside a seed tend to form at a particular angle to each other, about 137.5 degrees. That angle was well-known to ancient mathematicians and artists. Whether anyone noticed it in flowers before Kepler is uncertain, but it had a name: The golden ratio. It was understood to be a fundamental pattern of beauty in nature and art.

The golden ratio underlies the structure of pine cones, sunflower seeds and shellfish, Leonardo da Vinci’s paintings and the Parthenon. It’s in the spiral arms of distant galaxies, too. Somehow your eye imbibes it, and in late summer it wheels and dances in your mind like a kaleidoscope of billions and billions of goldenrods, sunflowers and other asters.

Dana Wilde lives in Troy. You can contact him at naturalist1@dwildepress.net. His new book “Winter: Notes and Numina from the Maine Woods” is available for pre-order from North Country Press. Backyard Naturalist appears the second and fourth Thursdays each month.