Katherine Hepburn
The Polka from 1844 sheet music.
The music is 2/4 time.
Although there is some dissension among Eastern European folklorists, most writing on origins of the dance indicate the name comes from the Czech word půlka for half, referring to the half-steps or hops dancers take.
I have not been able to find the actual reference. The closest thing might be this July, 1857 plate showing children's clothing in July, but that is a bonnet not a scarf.
Polka spots also found in an 1849 reference when the Richmond, Virginia's Southern & Western Literary Messenger's fictional Edgar was looking to improve his wardrobe. The tailor dressed him in a "Cravat, Polka spot— vest, white ground, sprinkled with gnats. Edgar looked handsome in his new suit."
An 1881 humorous essay from the Columbus, Ohio Capital
"From whence comes the polka-dot...?"
Gloria Swanson
But no one can tell us how polka became the name for a dotted pattern other than the generalization that Polka-mania inspired terms for fashion like Polka Skirts. Women dancing the Polka sometimes wore specific dress inspired by folk costumes.
1850s by Franz Antoine
The conventional wisdom on the internet is that the earliest printed reference for Polka Dot is said to be an 1857 caption in Godey's Lady's Book, a “Scarf of muslin, for light summer wear, surrounded by a scalloped edge, embroidered in rows of round polka dots.”
Never mind, the internet is wrong. 1857 is not the earliest printed reference. A fifteen-minute search through Newspapers.com reveals several earlier advertisements.
In September, 1850 the ladies of Charleston, South
Carolina could find "New Styles Polka Dot Muslin De Lains"
at Ketchum & Taylor.
Perhaps something like this wool/cotton combination purple
dot in a later log cabin block.
Delaine was a printed, mixed-wool fabric.
Over a year earlier one could buy similar fabric in silk in New Orleans.
This ad from April 21, 1849 advertises
"Changeable Polka Spot Silks"
and "Very Rich Polka-spot Foulard Silks"
Unknown couple, 1861-1865
A foulard print meant a specific type of design repeat,
a simple figure dropped half way as it progressed across the surface,
creating a diagonal pattern.
Polka spots & polka dots.
Alice B Haven's fiction in Godey's in 1859:
Mrs. Archer ...said: “Will you show me something for a child's dress medium colors for fall wear, a polka spot, or very small chintz figure on a plain ground."
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Mrs. Archer ...said: “Will you show me something for a child's dress medium colors for fall wear, a polka spot, or very small chintz figure on a plain ground."

The half-drop repeat on the left is a Polka Dot.
The full-drop repeat on the right is a dotted fabric.
Some may call that full-drop dot a Polka Dot but I have my standards.
Polka dots on the left. Bubbles perhaps on the right.
The key to the term polka dot may lie in the word půlka for half.
A half-drop repeat like the little jump in a polka dance.
From the Rocky Mountain Quilt Shop
Girl and dog about 1850
Library of Congress
I made up that last part about the half drop and half půlka but it sounds good to me.
I have had a lot of time to think lately.