In a very interesting essay by Robert Hillenbrand studying the works of Jila Peacock, the author gives an introduction about the development of Arabic calligraphy that I would like to summarize here:
It could be argued that calligraphy has been less subject to... the dominance of ideas from outside the Islamic world than have all its sister arts, from architecture to painting, from pottery to carpets. And the main reason is the high regard that calligraphy is held throughout the Islamic world and its association with the Qur'an. The flexibility and curvilinearity of the Arabic script can employ symmetry, echo, antithesis and other devices which can develop in any direction. This is aided by the special features of the Arabic script which include the interplay between angular and curved letters, capacity to handle compression and prolongation equally, the absence of capital letters producing visual unity, ease of adaptation to large and small scales, accommodation of shifts from the baseline to upper register, as well as apparent characteristics of dynamism, energy and rhythm.
The delight of constantly expanding the boundaries of expression explains why so many contrived scripts were invented in the course of centuries... this began with purely abstract forms and patterns created by careful placing of individual letters or blocks of text. These attempts soon lead to more radical experiments with calligraphy... which involved figural designs or zoomorphic elements. The earliest attempts to do this is often debated but it is more likely to have been originated somewhere near the eastern Iranian world in the twelfth century. This tendency started with incising faces onto the thickened upper shafts of the tallest letters... these faces could be an eyebrow, an eye and a mouth with a neck below... now the shafts of the taller letters have morphed into legless bodies, they interact with unmistakable humor... patting each other on the shoulder... shaking hands... Thus the upper storey of the inscription is shot through with narrative while its ground floor spells out a message of good wishes and happy life.
The next step in the practice of bringing living creatures into calligraphy is related to a loosening of earlier restraints and seems to have originated in the Iranian world. This new mode was not a matter of script metamorphosing into living forms which are also readable letters, but of using script to delineate such forms. This practice established itself only relatively late in Islamic art, when the taboos outlawing religious iconography had lost some of their power... It developed in... Ottoman Turkey, India and Qajar Iran... [and] was known as early as 1458.
An example scanned from the book "The splendor of Islamic Calligraphy" showing an early fusion of calligraphy and floral decoration that soon lead to more radical experimentation.
A traditional example of zoomorphic calligraphy by Hassan Musa.
The first verse of Surah Al-Feel (The elephant, 105) which translates to: Have you not considered how your Lord dealt with the possessors of the elephant?
A hoopoe by Jila Peacock formed by one of Hafez poems, a Persian poet whose statue in Iranian literature is equivalent to Shakespeare in English literature.
Hillenbrand ends his introduction to the context of calligraphy with these comments that I thought were spot-on:
In many of these calligraphic images, one senses that writing is being pushed to its furthest limits so as to make it express unnaturally what it cannot do naturally... These arcane images offer striking evidence as to the slightly perverse outlets into which Islamic artists channeled their frustrated desires to create religious pictures. It was a current which proved too strong for Islamic orthodoxy to dam.
Source for essay link and more info can be found here