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The positive meanings of the symbol were subverted in the early twentieth century when it was adopted as the emblem of the National Socialist German Workers Party.This association occurred because Nazism stated that the historical Aryans were the modern Germans and then proposed that, because of this, the subjugation of the world by Germany was desirable, and even predestined. Geometrically, the swastika can be regarded as an irregular icosagon or 20-sided polygon.Before this, it was particularly well-recognized in Europe from the archaeological work of Heinrich Schliemann, who discovered the symbol in the site of ancient Troy and who associated it with the ancient migrations of Indo-European (Aryan) peoples.

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Sagan suggests that in antiquity a comet could have approached so close to Earth that the jets of gas streaming from it, bent by the comet's rotation, became visible, leading to the adoption of the swastika as a symbol across the world.Theories of single origin as a sacred prehistorical symbol point to the Proto-Indo-Europeans, noting that the swastika was not adopted by Sumer in Mesopotamia, which was established no later than 3500 BC, and the Old Kingdom of Egypt, beginning in 2630 BC, arguing that these were already well-established and codified at the time of the symbol's diffusion.The swastika has not always been used as a symbol of Nazism and was in fact borrowed from Eastern cultures.It seems to have first been used by early inhabitants of Eurasia.The swastika remains a core symbol of Neo-Nazi groups.

Since the end of World War II, the traditional uses of swastika in the western world were discouraged.Schliemann concluded that the swastika was a specifically Aryan symbol.This idea was taken up by many other writers, and the swastika quickly became popular in the West, appearing in many designs from the 1880s to the 1920s.The swastika is chiral, with no reflectional symmetry, but both mirror-image forms have 90 rotational symmetry (that is, the symmetry of the cyclic group C4).Looking at an upright swastika, the upper arm clearly faces towards the viewer's left (SM) or right (SP).The first theory is that the swastika's symmetry and simplicity led to its independent development everywhere, along the lines of Carl Jung's collective unconscious, or just as a very simple symbol.