Regular Figures and Circular Order

Elements

This flow follows Book IV's programme of inscribing and circumscribing regular figures around circles.

increasing

Bearing strength in each event

decreasing

Inscribed Triangle Inscribed Triangle from "Elements". Inscribed Triangle Angular Transfer A triangle is inscribed in a given circle so that its angles correspond to those of a given triangle.

Inscribed Triangle from "Elements". Inscribed Triangle Angular Transfer A triangle is inscribed in a given circle so that its angles correspond to those of a given triangle.

Inscribed Triangle Angular Transfer A triangle is inscribed in a given circle so that its angles correspond to those of a given triangle. Tangents and alternate segment relations transfer angular information to the circumference. Circular Realization A prescribed shape becomes possible through the circle's boundary.

Circumscribed Triangle Circumscribed Triangle from "Elements". Circumscribed Triangle Tangent Sides Tangents drawn at selected radii form a triangle around a circle.

Circumscribed Triangle from "Elements". Circumscribed Triangle Tangent Sides Tangents drawn at selected radii form a triangle around a circle.

Circumscribed Triangle Tangent Sides Tangents drawn at selected radii form a triangle around a circle. The proof compares the new angles with those of a given triangle. External Form The circle becomes an internal organizing centre for a larger rectilinear figure.

Square Construction Square Construction from "Elements". Square Construction Fourfold Symmetry Perpendicular diameters create four equal arcs and right angles.

Square Construction from "Elements". Square Construction Fourfold Symmetry Perpendicular diameters create four equal arcs and right angles.

Square Construction Fourfold Symmetry Perpendicular diameters create four equal arcs and right angles. Joining their endpoints produces an inscribed square, while tangents produce a circumscribed one. Circular Order The construction shows how regularity can be generated from the simplest right-angle relations.

Pentagon Construction Pentagon Construction from "Elements". Pentagon Construction Special Isosceles Triangle An isosceles triangle with a doubled base angle supplies the arc division required for a regular pentagon.

Pentagon Construction from "Elements". Pentagon Construction Special Isosceles Triangle An isosceles triangle with a doubled base angle supplies the arc division required for a regular pentagon.

Pentagon Construction Special Isosceles Triangle An isosceles triangle with a doubled base angle supplies the arc division required for a regular pentagon. The triangle is inscribed and its base angles are bisected. Regularity Equal arcs produce equal chords and equal angles, completing the pentagonal figure.

Hexagon and Quindecagon Hexagon and Quindecagon from "Elements". Hexagon and Quindecagon Sixfold Division An equilateral triangle and the circle's radius generate the regular hexagon.

Hexagon and Quindecagon from "Elements". Hexagon and Quindecagon Sixfold Division An equilateral triangle and the circle's radius generate the regular hexagon.

Hexagon and Quindecagon Sixfold Division An equilateral triangle and the circle's radius generate the regular hexagon. The construction depends on equal radii and equal central angles. Fifteenfold Division The pentagon and triangle are combined to obtain the quindecagon, extending circular order through arc subtraction.

at the second proposition of Book IV

at the third proposition of Book IV

at the sixth and seventh propositions of Book IV

at the eleventh and twelfth propositions of Book IV

at the fifteenth and sixteenth propositions of Book IV