A sesqui-quadrate, also called the sesquisquare or trioctile, is the aspect of one hundred and thirty-five degrees, a square plus a semi-square. It is one of the minor aspects that research into Johannes Kepler's astrological writing credits to him by name, introduced in the Harmonices Mundi of 1619 alongside the quintile and biquintile as part of his argument that workable aspect angles ought to follow from the geometry of the regular polygons that can be constructed with compass and straightedge, and from the musical consonances those polygons were held to embody, rather than from the sign-based scheme inherited from earlier astrology.
Being three-eighths of the circle it belongs, geometrically, with the semi-square to the family of aspects produced by repeated halving, even though its authorship is more securely Kepler's own than the semi-square's is generally said to be. It is read as a hard minor aspect of the same character as the semi-square, agitation and recurring provocation rather than a settled block, and like the other minor aspects it is customarily given a narrow orb. The alternative name trioctile places it in the same eighth-part family by number as the semi-square, one eighth, and the square, two eighths, so that all three hard aspects of that family, semi-square, square and sesqui-quadrate, can be read off a single progression of eighths of the circle: one, two and three.