What does the symbol of a crescent moon with three stars mean?

There is a good deal of discussion on this subject so what I'll do is list some of the theories for you so you can see how different cultures and faiths have claimed this symbol for their own. As with many symbols (including the cross, the swastika and the pentacle) the symbol has come to denote different meanings and over time the meanings have become so interchangeable that no single absolute answer would be possible.

1. The symbol is possibly most recognisable (in both this form and with a single star) as a symbol of the Islamic faith. Islam, a world religion originating in the East has strong links to the moon cycle and the most important phase of the Muslim calendar - the fasting month of Ramadan - is determined by the cycle of the moon. Islam is not a Pagan religion but an Abrahamic religion - which alongside Christianity evolved from the parent religion of Judaism. You may notice that a similar image (usually a single star) appears on the flags of many countries, this is to show their faith in Islam.

2. The crescent moon and stars first appear in the carvings left to us by the Sumerian culture - possibly the oldest civilisation that ever lived on our planet. The Sumerians were said to have a knowledge of the solar system that far surpassed anything their primitive technology should have allowed them to know and they left this knowledge to us in the form of carvings and sculpture. The moon and stars represented the Gods to the Sumerians and would have been considered a symbol of worship. There is a school of thought that when the image appeared as a single star it was representative of Venus which is the second brightest object in the night sky after the moon.

3. The symbol occurs throughout history as being symbolic of a specific God or (more commonly) Goddess. Such associations include the Greek hunting Goddess Artemis (Diana in Roman mythology), the Greek Goddess of the night Nyx, Tanit the Great Goddess of the Carthaginians and Astarte (Ishtar) the Mesopotamean (Eastern) Goddess associated with War. 

4. The Ottoman Empire (Turkey and the surrounding areas) of the middle ages observed a religion called Tengrism. This religion was known as the "Sky God religion" and great emphasis was placed upon using symbols of the skies to enhance worship. The Moon and stars were depicted as symbols on the Ottoman flags and this flag morphed over time to become the crescent moon and single star depicted on the flag of Turkey as we know it today.

5. Some modern Pagans do recognise the symbol for their own uses - the significance of the moon has obvious Pagan connotations, the points on the stars represent the five elements associated with the elements and the number of stars represent the transition of the Goddess from Maiden through Mother to Crone. The image of the moon and stars also evokes the idea of perfect balance - our planet as we know it is a balance of night and day, dark and light, heat and cold and this balance can be symbolised by the image of the moon and the star (the sun) that we see in our own Earth skies. The idea of the Moon Goddess and the Sun God in modern Pagan religions such as Wicca can also be encapsulated in this image.

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