'Carmelite' sounds a bit geological, like 'stalagmite', a column of calcium carbonate one might see in a limestone cave. But it isn't.
And a 'Discalced Carmelite' sounds like the calcium has been removed, but believe me, that is not the case.
A Discalced Carmelite just means a friar of the Carmelite Order who traditionally walks around barefoot, and their bodies do contain the normal (1.5%) amount of calcium. Yet there actually is a geological link to the Carmelites; tenuous, but rock solid.
Their full name is the 'Order of the Brothers of Our Lady of Mount Carmel' (Ordo fratrum Beatæ Virginis Mariæ de monte Carmelo). The mountain range called Carmel in Northern Israel actually contains many limestone caves, and it is within one of these caves that the prophet Elijah is believed to have lived.
Whether Elijah was following a trend of seclusion on the mountain or whether he set that trend, by the 12th century several hermits lived on Carmel.
A recluse lives apart from the distractions of the main society. That doesn't mean the recluse lives in isolation; quite often we find small communities living away from the main society. One such group was the original Carmelites, and their current emblem (shown above) features a mountain peak in the centre to symbolise Mount Carmel.
This emblem is now used by many different Carmelite groups and almost invariably includes representation of a mountain and three stars. The number of points on each star is not consistent between the groups, with five or six points being the most common.
5-pointed stars. Discalced Carmelite Order
6-pointed stars. Carmelite Sisters of Los Angeles
8-pointed stars. Roman Discalced Carmelites
As for the meaning of the three stars, they could represent the Holy Trinity but usually they don't. One interpretation is that the upper stars represent Elijah and Elisha, patriarchs of the Order, and the lower star represents the Blessed Virgin Mary. Carmelites are a Marian Order and the star at the foot of the cross represents Mary as Queen of Carmel. The position of that star on the emblem is to show that Mary is at the heart of the Church.
A different interpretation, however, is from Alex Roman who tells us that...
...three stars are often found on Eastern Orthodox icons of the Blessed Virgin Mary - on her shoulders and on her forehead. Those three stars signify that she was a Virgin "Before, During and After" the Birth of Christ. Given the strong Marian character of the Carmelites, it is truly an appropriate symbolism. Mary herself is often referred to liturgically as the "Uncut Mountain" and also relates to an Old Testament prophecy that the Messiah will come "from a Mountain, densely wooded".
The fact that the middle star is at the base of Mount Carmel also gave rise to the title of the Virgin Mary "Star of the Sea" who is invoked by those travelling on water.
The brown colour not only relates to the earth (of the mountain), but also to the skin colour of the original icon of Our Lady of Mount Carmel that the Carmelites brought to Rome from the Holy Land and which is now kept in the Cathedral of Naples, Italy. She is called "La Bruna" owing to her brown, dark skin colour and this recalls the first chapter of the Song of Songs "For the sun gave me a tan" (Song. 1:6). Here the "sun" is Christ Himself.
The further reference there to the vineyard represents the Church and is also an appropriate symbol for Carmel which means "garden" in Hebrew.
Some have adopted a logo without the cross. Why they have chosen to omit an obvious Christian identity is best known to themselves.
It may look a bit like a back pocket on a pair of denim jeans, but since the emblem belongs to those communities, it is not for outsiders to question their choice. (Some Christians choose not to wear a Christian I.D. badge, preferring their lives to be their witness. Others feel that including a cross on these ubiquitous emblems is demeaning to such an important sacrament.)
The Order evolved, as many things do, and over the centuries became nothing like the original. In the 16th century, one of several reforms saw a return to a more ascetic and austere way of life, to the point where the friars walked around barefoot.
The Latin for shoe is calce and the friars who followed this reform were called Discalced Carmelites.
Elijah is perhaps best known for his challenge to the Pagan worshippers of Baal gods.
It was a competition of sorts, where Baal priests built an altar to roast a sacrifice. Elijah did the same for the God of Israel, but rather than light the fires with flints, the priests prayed for many hours to Baal to miraculously strike the tinder with lightning. Sadly for them, no lightning appeared. They even slashed their bodies to add their own blood to the altar, but still the fire wasn't lit.
When they finally gave up, it was Elijah's turn. But before Elijah prayed to his God, he drenched his altar with water. Then he prayed, the lightning struck, and the point was made. (See 1 Kings 18:20-40)