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The most common regular polygon in Islamic art design is the octagon. Historical evidence of the use of an 8-star polygon and an 8-fold rosette dates back to Jemdet Nasr (3100-2900 B.C.) in Mesopotamia. Additionally, in ancient Egypt, octagons can be found in mathematical problem (Ahmose papyrus, Problem number 48), household goods (papyrus storage), architecture (granite columns) and decorations (palace decorations). The regular octagon which is a fundamentally important element of Islamic art design, is widely used as arithmetic objects in metric algebra along with other regular polygons in Mesopotamia. The 8-point star polygon has long been a symbol of the ancient Sumerian goddess Inanna and her East Semitic counterpart Ishtar. During the Neo-Assyrian period, the 8-fold rosette occasionally replaced the star as the symbol of Ishtar. In this paper, we discuss how octagonal design prevailed in the Islamic region since the late ninth century, and has existed in Mesopotamia from Jemdet Nasr to the end of third century B.C. We describe reasons why the geometric pattern of regular polygons, including regular octagons, developed in the Islamic world. Furthermore, we also discuss mathematical meanings of regular polygons.

In this paper, we define and study extensively a new type of external sets in , we call it "convex galaxies". We show that these convex external sets may be classified in some definite types. More precisely, we obtain the following : (1)Let be a convex galaxy which is symmetric with respect to zero, then (i) is an - galaxy (0) if and only if there exists an internal strictly increasing sequence of strictly positive real numbers with such that and , for all , where, is some limited real number such that . (ii) is a non-linear galaxy if and only if there exists an internal strictly increasing sequence of strictly positive real numbers with such that is unlimited for all . (2)Let be a convex galaxy which is symmetric with respect to zero, then (i) is an - galaxy (0) iff there exists a real internal strictly increasing - function , such that , and for all limited , where is a positive real number. (ii) is a non-linear galaxy if and only if there exists a real internal strictly increasing - function , such that and is positive unlimited, for all appreciable .