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Shlokam:

In the form of male and female, matted hair with moon, axe, smeared in red (aruNa), wearing snakes and the skin of tiger, embracing the bull and with a bent leg, this the right half, and the left black in color, hand holding a (lily) flower close to the chest, with a silk like lotus foot, wearing gold ornaments, salutations to the Half female God.

purANa of the deity

ardha nAri Ishvarar : half female God (bhOga mUrti)

Sage bRingi is one of the ardent devotees of Lord shiva. He used to worship only Lord shiva and not shakti. Goddess shakti, being the power as the name indicate, pulled out the energy from bRingi mahaRishi's body. Now he was even unable to stand. He pleaded to God. God shiva provided him with a stick. On its support he stood and still worshipped Lord shiva alone. Goddess shakti wanted to become an inseparable part of Lord shiva's form. She oberved the kEdhAra mahA vrata (1) austerity, which is now known as deepAvaLi. Pleased with her austerity, Lord shiva granted her the boon of being part of His form. So the Lord now appeared male on the right side and female on the left side and hence became ardhanArIshvara.

 

  

Significance of this form

This is one of the very important form of God, Hindus worship. This is a much-hailed form in scriptures of various languages. Hindus do not say that the God is only male. God is male - female and neuter too ! Since God is conceptually beyond sex, though gets referred as He/She many times, it is more appropriate to refer as It, especially at Its intrinsic condition, as do many hindu scriptures.

Philosophically, this form is quite associated with the Grace of God. shiva and shakti are one and the same Supreme. The formless God is called parashiva. On Its own free-will for the benefit of pashus(souls), which are drowned in pAsha (bondage), It thinks to create the worlds. Its dynamism of creation thus springs out of It, which is called shakti. Now shiva and Its power shakti create everything. This is the form of their togetherness that springs out of Lord shiva. Because of this the form is associated with the grace of God. shiva and shakti though the same may also act independently. They are associated like the person and the action of the person. They are one and the same like the ice and the water - one becomes the other. For this reason scriptures describe that shakti to shiva is a wife (they are together), mother (shakti becomes shiva - so shiva comes out of shakti) and daughter (shiva become shakti). Poet kAlidAsa hails them as inseparable like the word and its meaning, the letter and pronunciation !!

See Also: 
1. kEdAra vratam 

1. விரிகதிர் ஞாயிறல்லர் மதியல்லர் வேத விதியல்லர் விண்ணு நிலனுந்

திரிதரு வாயுவல்லர் செறுதீயு மல்லர் தெளிநீரு மல்லர் தெரியில்

அரிதரு கண்ணியாளை ஒருபாக மாக அருள்கார ணத்தில் வருவார்

எரியர வாரமார்பர் இமையாரு மல்லர் இமைப்பாரு மல்லர் இவரே.  4.8.2

 

2. தோலுந் துகிலுங் குழையுஞ் சுருள்தோடும்

பால்வெள்ளை நீறும் பசுஞ்சாந்தும் பைங்கிளியுஞ்

சூலமுந் தொக்க வளையு முடைத்தொன்மைக்

கோலமே நோக்கிக் குளிர்ந்தூதாய் கோத்தும்பீ. - திருவாசகம்