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Symmetry in Medieval heraldry is not the same as the modern notion of line or mirror image symmetry. Rather, two or more of the same object, or charge, placed on either side of a line of symmetry must be facing in the same direction. The line of symmetry may be vertical, horizontal, or diagonal. Thus, symmetry consists of balance rather than mirroring in the eye of the Medieval heraldry designer.
Each of the following heraldric designs are symmetric in the Medieval sense. The first have a vertical line of symmetry, the third has both a horizontal and vertical line of symmetry and the last has a diagonal line of symmetry.
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This overall notion of symmetry as balance rather than mirror image comes from the Medieval world view that change is undesirable, that the world is balance, that it is and should be fixed and static with God in his Heaven and the every person in their proper place.
Reference
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