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Higher dimensions Orthogonal projections of higher dimensional polytopes can also create pentagrammic figures: 4D 5D 4-simplex t0.svg The regular 5-cell (4-simplex) has five vertices and 10 edges. 4-simplex t1.svg The rectified 5-cell has 10 vertices and 30 edges. 5-simplex t1 A4.svg The rectified 5-simplex has 15 vertices, seen in this orthogonal projection as three nested pentagrams. 5-simplex t2 A4.svg The birectified 5-simplex has 20 vertices, seen in this orthogonal projection as four overlapping pentagrams. All ten 4-dimensional Schläfli–Hess 4-polytopes have either pentagrammic faces or vertex figure elements. Pentagram of Venus The pentagram of Venus The pentagram of Venus is the apparent path of the planet Venus as observed from Earth. Successive inferior conjunctions of Venus repeat with an orbital resonance of approximately 13:8—that is, Venus orbits the Sun approximately 13 times for every eight orbits of Earth—shifting 144° at each inferior conjunction.[34] The tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram, and each group of five intersections equidistant from the figure's center have the same geometric relationship
The Seal of Solomon (or Ring of Solomon; Arabic: ???? ??????? Khatam Sulayman) is the signet ring attributed to King Solomon in medieval Jewish tradition and in Islamic and Western occultism. It was often depicted in either a pentagram or hexagram shape; the latter also known as the Star of David in Jewish tradition. This ring variously gave Solomon the power to command demons, jinn (genies) and spirits, or to speak with animals. Due to the proverbial wisdom of Solomon, his signet ring, or its supposed design, it came to be seen as an amulet or talisman, or a symbol or character in medieval and Renaissance-era magic, occultism, and alchemy.



quasitruncation, produces a decagram, {10/3}. Regular star truncation 5-3 2.svg shallow t{5/2} Regular star truncation 5-3 1.svg t{5/3} = {10/3} Golden ratio A regular pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another. The golden ratio, f = (1 + v5) / 2 ˜ 1.618, satisfying {\displaystyle \varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }\,}\varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }\, {\displaystyle \varphi =1/(2\sin(\pi /10))=1/(2\sin 18^{\circ })\,}\varphi =1/(2\sin(\pi /10))=1/(2\sin 18^{\circ })\, {\displaystyle \varphi =2\cos(\pi /5)=2\cos 36^{\circ }\,}\varphi =2\cos(\pi /5)=2\cos 36^{\circ }\, plays an important role in regular pentagons and pentagrams. Each intersection of edges sections the edges in the golden ratio: the ratio of the length of the edge to the longer segment is f, as is the length of the longer segment to the shorter. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is f. As the four-color illustration shows: {\displaystyle {\frac {\mathrm {red} }{\mathrm {green} }}={\frac {\mathrm {green} }{\mathrm {blue} }}={\frac {\mathrm {blue} }{\mathrm {magenta} }}=\varphi .}{\frac {{\mathrm {red}}}{{\mathrm {green}}}}={\frac {{\mathrm {green}}}{{\mathrm {blue}}}}={\frac {{\mathrm {blue}}}{{\mathrm {magenta}}}}=\varphi . The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is f. The acute triangles are golden triangles. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon. Trigonometric values Further information: Trigonometric constants expressed in real radicals: 36°: regular pentagon {\displaystyle {\begin{aligned}\sin {\frac {\pi }{10}}&=\sin 18^{\circ }={\frac {{\sqrt {5}}-1}{4}}={\frac {\varphi -1}{2}}={\frac {1}{2\varphi }}\\[5pt]\cos {\frac {\pi }{10}}&=\cos 18^{\circ }={\frac {\sqrt {2(5+{\sqrt {5}})}}{4}}\\[5pt]\tan {\frac {\pi }{10}}&=\tan 18^{\circ }={\frac {\sqrt {5(5-2{\sqrt {5}})}}{5}}\\[5pt]\cot {\frac {\pi }{10}}&=\cot 18^{\circ }={\sqrt {5+2{\sqrt {5}}}}\\[5pt]\sin {\frac {\pi }{5}}&=\sin 36^{\circ }={\frac {\sqrt {2(5-{\sqrt {5}})}}{4}}\\[5pt]\cos {\frac {\pi }{5}}&=\cos 36^{\circ }={\frac {{\sqrt {5}}+1}{4}}={\frac {\varphi }{2}}\\[5pt]\tan {\frac {\pi }{5}}&=\tan 36^{\circ }={\sqrt {5-2{\sqrt {5}}}}\\[5pt]\cot {\frac {\pi }{5}}&=\cot 36^{\circ }={\frac {\sqrt {5(5+2{\sqrt {5}})}}{5}}\end{aligned}}}{\displaystyle {\begin{aligned}\sin {\frac {\pi }{10}}&=\sin 18^{\circ }={\frac {{\sqrt {5}}-1}{4}}={\frac {\varphi -1}{2}}={\frac {1}{2\varphi }}\\[5pt]\cos {\frac {\pi }{10}}&=\cos 18^{\circ }={\frac {\sqrt {2(5+{\sqrt {5}})}}{4}}\\[5pt]\tan {\frac {\pi }{10}}&=\tan 18^{\circ }={\frac {\sqrt {5(5-2{\sqrt {5}})}}{5}}\\[5pt]\cot {\frac {\pi }{10}}&=\cot 18^{\circ }={\sqrt {5+2{\sqrt {5}}}}\\[5pt]\sin {\frac {\pi }{5}}&=\sin 36^{\circ }={\frac {\sqrt {2(5-{\sqrt {5}})}}{4}}\\[5pt]\cos {\frac {\pi }{5}}&=\cos 36^{\circ }={\frac {{\sqrt {5}}+1}{4}}={\frac {\varphi }{2}}\\[5pt]\tan {\frac {\pi }{5}}&=\tan 36^{\circ }={\sqrt {5-2{\sqrt {5}}}}\\[5pt]\cot {\frac {\pi }{5}}&=\cot 36^{\circ }={\frac {\sqrt {5(5+2{\sqrt {5}})}}{5}}\end{aligned}}} As a result, in an isosceles triangle with one or two angles of 36°, the longer of the two side lengths is f times that of the shorter of the two, both in the case of the acute as in the case of the obtuse triangle. Spherical pentagram Further information: Pentagramma mirificum A pentagram can be drawn as a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). It was studied later by Carl Friedrich Gauss. Three-dimensional figures Further information: Uniform polyhedron: Icosahedral symmetry Several polyhedra incorporate pentagrams: Pentagrammic prism Pentagrammic antiprism Pentagrammic crossed-antiprism Small stellated dodecahedron Great stellated dodecahedron Small ditrigonal icosidodeca­hedron Dodecadodecahedron The legend of the Seal of Solomon was developed primarily by medieval Arabic writers, who related that the ring was engraved by God and was given to the king directly from heaven. The ring was made from brass and iron, and the two parts were used to seal written commands to good and evil spirits, respectively. In one tale, a demon—either Asmodeus or Sakhr—obtained possession of the ring and ruled in Solomon's stead for forty days. In a variant of the tale of the ring of Polycrates from Herodotus, the demon eventually threw the ring into the sea, where it was swallowed by a fish, caught by a fisherman, and served to Solomon.[1] In Islamic eschatology, the Beast of the Earth is equipped with both the Staff of Moses and the Seal of Solomon and uses the latter to stamp the nose of the unbelievers.[2] The date of origin legends surrounding the Seal of Solomon is difficult to establish. It is known that a legend of a magic ring with which the possessor could command demons was already current in the 1st century (Josephus 8.2 telling of one Eleazar who used such a ring in the presence of Vespasian), but the association of the name of Solomon with such a ring is medieval notwithstanding the 2nd century apocryphal text the Testament of Solomon. The Tractate Gittin (fol. 68) of the Talmud has a story involving Solomon, Asmodeus, and a ring with the divine name engraved.[3] The specification of the design of the seal as a [hexagram] seems to arise from a medieval Arab tradition. The name "Solomon's seal" was given to the hexagram engraved on the bottom of drinking-cups in Arab tradition. In the Arabian Nights (chapter 20), Sindbad presented Harun al-Rashid with such a cup, on which the "Table of Solomon" was engraved.[4] Hexagrams feature prominently in Jewish esoteric literature from the early medieval period, and some authors have hypothesized that the tradition of Solomon's Seal may possibly predate Islam and date to early Rabbinical esoteric tradition, or too early alchemy in Hellenistic Judaism in 3rd-century Egypt, but there is no positive evidence for this, and most scholars assume that the symbol entered the Kabbalistic tradition of medieval Spain from Arabic literature.[5] The representation as a pentagram, by contrast, seems to arise in the Western tradition of Renaissance magic (which was in turn strongly influenced by medieval Arab and Jewish occultism); White Kennett (1660–1728) makes reference to a "pentangle of Solomon" with the power of exorcising demons.[6] The hexagram or "Star of David", which became a symbol of Judaism in the modern period and was placed on the flag of Israel in 1948, has its origins in 14th-century depictions of the Seal of Solomon. In 1354, King of Bohemia Charles IV prescribed for the Jews of Prague a red flag with both David's shield and Solomon's seal, while the red flag with which the Jews met King Matthias of Hungary in the 15th century showed two pentagrams with two golden stars.[7] Peter de Abano's Heptameron (1496) makes reference to the "Pentacle of Solomon" (actually a hexagram is drawn on the floor in which the magician has to stand) to invoke various demons.[8] Lippmann Moses Büschenthal (d. 1818) wrote a tragedy with the title Der Siegelring Salomonis ("the signet-ring of Solomon"). An "Order of the Seal of Solomon" was established in 1874 in Ethiopia, where the ruling house claimed descent from Solomon. What distinguishes a Seal of Solomon from a Star of David is the two triangles are interlaced giving the appearance of a 3-dimensional figure. This was said in the Testament of Solomon to make demons confused and dizzy, unable to do Solomon any harm


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