The smile, it seems, is not just a product of Carroll's imagination, but a part of human culture. We remember how the wife of 'Mercury' (The Mad Hatter) according to Inuit myth tried to make the victims smile:

 ... Finally, at one time, he could really be heard to enter to them, he, the poor cousin of the Moon, the entrail-snatcher, carrying a dish and a large knife, in order to try to snatch the entrails of the human being. And look! At the window his wife stood and kept on saying: 'She smiles!' The entrail-snatcher began to dance a drum dance, with ridiculous movements, and they only looked at him, while he sang: My little dogs, I get them food, / My little dogs, I get them food, / ha-ahing, ha-ahing, ha-ahing. While he acted thus, his poor wife all along stood at the window saying: 'She smiles, she smiles, she smiles!' ...

To smile means to be like the sun. The equality would enable 'the poor cousin' to use his knife. I identify the Mad Hatter with Mercury not only because of his position (at autumn - where water falls from heaven, running downwards like the heavy mercury liquid), but also because of Carroll's decision to include the character at the T table:

"The '10/6' on the Hatter's hat means ten shillings and six pence, the price of the hat in pre-decimalised British money and acts as a visual indication of the hatters trade. The name 'Mad Hatter' was undoubtedly inspired by the phrase 'as mad as a hatter'. There is some confusion as to the origins of this phrase, however there is scientific evidence behind the link between the hatters and insanity. Mercury was used in the process of curing felt used in some hats. It was impossible for hatters to avoid inhaling the mercury fumes given off during the hat making process. Hatters and mill workers often suffered mercury poisoning as residual mercury caused neurological damage including confused speech and distorted vision. It was not unusual then for hatters to appear disturbed or mentally confused, many died early as a result of mercury poisoning ..." (Wikipedia)

A gnomon is formed like an L:

J maybe was inserted to indicate the turnover due with the 10th position? T (as in Twenty) makes us remember the fate of Hercules:

... The manner of his death can be reconstructed from a variety of legends, folk-customs and other religious survivals. At mid-summer, at the end of a half-year reign, Hercules is made drunk with mead and led into the middle of a circle of twelve stones arranged around an oak, in front of which stands an altar-stone; the oak has been lopped until it is T-shaped. He is bound to it with willow thongs in the 'five-fold bond' which joins wrists, neck, and ankles together, beaten by his comrades till he faints, then flayed, blinded, castrated, impaled with a mistletoe stake, and finally hacked into joints on the altar-stone. His blood is caught in a basin and used for sprinkling the whole tribe to make them vigorous and fruitful. The joints are roasted at twin fires of oak-loppings, kindled with sacred fire preserved from a lightning-blasted oak or made by twirling an alder- or cornel-wood fire-drill in an oak log.

The trunk is then uprooted and split into faggots which are added to the flames. The twelve merry-men rush in a wild figure-of-eight dance around the fires, singing ecstatically and tearing at the flesh with their teeth. The bloody remains are burnt in the fire, all except the genitals and the head. These are put into an alder-wood boat and floated down the river to an islet; though the head is sometimes cured with smoke and preserved for oracular use. His tanist succeeds him and reigns for the remainder of the year, when he is sacrificially killed by a new Hercules ...

His head was sometimes used for oracular purposes, just as the turtle shell was used for predicting the future in ancient China. During the dark (tanist) season the future will be determined.