Foxtrot in the Great Gatsby, 1925

A Foxtrot from 1926

 "I'd a little rather not be the polo player," said Tom pleasantly, "I'd rather look at all these famous people in--in oblivion."
Daisy and Gatsby danced. I remember being surprised by his graceful, conservative fox-trot--I had never seen him dance before. Then they sauntered over to my house and sat on the steps for half an hour while at her request I remained watchfully in the garden: "In case there's a fire or a flood," she explained, "or any act of God."
Tom appeared from his oblivion as we were sitting down to supper together. "Do you mind if I eat with some people over here?" he said. "A fellow's getting off some funny stuff." (F.Scott Fitzgerald, The Great Gatsby)

In contrast to the wild extravagance of a shimmy dancer, Gatsby and Daisy know the current elegant dances. The foxtrot was new and popular in the 1920s and, according to Coll and Rossiere, it was especially dignified.
 
"Fox Trot is hardly a descriptive title, as indeed this dance, which seems to hold the fundamental principles of modern dances, is most unlike a trot. Indeed it is a smooth, easy dance and the most dignified of modern dances." ( Coll, Charles J. & Gabrielle Rosiere Dancing made easy. New and rev. ed.  New York: E. J. Clode 1922. p. 73.)

The easiest way to approach the foxtrot is to think of it as a combination of one-step (slow) and two-step (quick). According to A.M. Cree, you can string steps together in any combination  you like. One pattern that we like to use is counted like this, 1, 2, 3, and 4, 5, 6, and 7, 8. or the same pattern can be counted like this: slow, slow, quick, quick, slow, slow, quick, quick, slow, slow. 

The fun thing about Foxtrot is that dance manuals of the time used diagrams to show patterns of steps on the floor. As a dancer, you can think about those geometrically inspired patterns made by your feet, even while the fact that you are dancing with a partner makes those patterns appear softer and indistinct to a watcher.

Diagram from Cree, 1920

THE Fox-Trot is an American importation that, I think, has come to stay. It is an admirable dance and has become very popular in England in the last few years, the movements being simple and suitable for a fast-moving and full ball-room, and the beat of the music being "catchy" and full of spirit.

The music is very similar to that of a One-Step, but slower, the beat being best thought of as "1-2" (one foot), "3-4" (other foot). It will be seen from the diagram that the dance is really a combination of movements--walking, running, gliding, and two-stepping--and not a distinct set of movements. The various movements should be performed at will, not in any given order and not any given number of times. (Cree, Aubrey McMahon, Handbook of ball-room dancing, London, John Lane; New York, John Lane Co., 1920. p. 51-52)

Diagram from Coll and Rossiere 1922

The name foxtrot comes from horses, not foxes, not someone named Fox. In the late 19th century, American horses had a variety of gaits. The main gaits were walk, trot and canter, but there were 4 other distinct gaits and foxtrot was one of those secondary gaits.

The walk, trot, pace, and gallop are the cardinal or natural gaits, and need no description to be generally understood. The artificial and acquired gaits are as follows:
First—The Fox Trot is a slight modification of the true trot, and perhaps to call it a 'broken' trot, in which the fore foot touches the ground an instant in advance of the diagonal hind foot, will convey as correct an understanding of it as a most elaborate description of the alternate placing of the feet. It has the slowest limits of the artificial gaits, and is an all-day gait. Its range is generally from four to six miles per hour, though occasionally a horse may show a rate of seven miles an hour. It is the utility gait of the general saddle horse, and every one worthy the name is expected to have it. At this gait a loose rein is always used, and the horse is apt to carry his head low. Drivers of trotting horses in harness will recognize it as a step frequently taken for two or three strokes by horses just in the act of slowing into a walk. (Wallace's Monthly, vol. 10, no 1, February 1884, New York: John H. Wallace, p. 15)

By saddle animal I do not mean what is known as a gaited saddle horse; although I do believe that if our cavalry horses were bred from fox-trotters, and our officers and men taught how to ride them, we could break all records for distance traveled in the field and for condition of both man and horse during and at the end of a campaign. The foxtrot does not in any way interfere with the regular walk, trot and gallop. It is merely a gait between the fast walk and the pounding trot, at which the horse can continue for almost as many hours as he can at a walk, and cover about one-third more distance. It is easy for the horse and for the man, and at the end of the day the horse will be comparatively fresh and the rider will not have had every bone in his body pounded until it is sore. He will be fit for something if occasion demands it in the way of guard or night work of any kind. 

Do not confuse the fox trot with the dogtrot, for they are as distinct as the trot and gallop. The fox-trot is peculiar to the United States, and a cavalry command mounted on fox-trotters could make its marches of thirty and thirty-five miles every day with less grief to horse and man than we can at present make twenty and twenty-five miles. Any saddle-horse man will confirm this statement. Such an ability would be a supreme advantage to an army. To secure it, however, would take several years' attention—probably ten or fifteen, and the experiment will not readily be made in this country. (Maj. Lloyd S. McCormick, Seventh Cavalry. Remounts; a plan for providing suitable horses for cavalry and artillery purposes. Armor, Volume 15, U.S. Army Armor School., 1905 - p. 587-588)

So, in terms of styling, we can think of a Foxtrot as an easy comfortable dance that won't tire out the dancers. Just in case all this talk of horses is confusing, you can see a lovely foxtrot on British Pathé by following this link: A Foxtrot 1924