The relevant congruences (shown when you move the "time" slider) are those between the central triangle formed by the vertices

,

,

and the three triangles

,

, and

. When the "time" slider begins to slide,

turns red, then rotates

radians about the axis through vertex

to become the central triangle

, then rotates

radians about the axis through vertex

to become

. Then

turns green and slides along the arc defined by the side

, showing that the two

radian rotations compose to a rotation about an axis normal to the great circle containing the arc

, but with the rotation angle twice the arclength of the arc

. Analogous relationships hold between

,

, and the arc

, as well as between

,

, and the arc

. The congruences shown by the moving triangles show that the rotation mapping

to

followed by the rotation mapping

to

is the rotation mapping

to

.

**Step 1. The Spherical Triangle Representing Quaternion Multiplication Alone**Click the checkboxes "show inner triangle only," "show vertex vectors," and "show labels." You see three unit vectors

,

,

defining vertices on the spherical triangle

. The directed arcs

,

, and

are the unit quaternions defined by the pairs

,

, and

of vectors, respectively. Actually, the arcs themselves contain one more bit of information than the pairs of vectors delimiting them: they define which of the two possible sections of the great circle marked out by the vectors is the arc in question.

The pair

uniquely defines the arc

. Indeed, given any two three-dimensional vectors

,

, one vector in the pair together with

and

uniquely determines the other vector; for example:

;

that is, one can invert the product defined by

, although of course neither the scalar nor vector product on its own is invertible. The pairing

is a standard way (see, e.g. [1], Chapter 1) to think of a quaternion; indeed one could say that the scalar and vector products of three-dimensional vectors from "modern" vector calculus are simply disembodied parts of the original quaternion product. With the geometry that we see in this Demonstration, it can be shown that the great circle arc

closing the spherical triangle comprising great circle arcs is indeed the quaternion product

.

The relationship between quaternion and rotation multiplication scaling can be intuitively understood by joining Roger Penrose [2] in his wonderful explanation of the triangle law for composition of rotations in §11.4 "How to Compose Rotations"; quaternions themselves are discussed more fully in the early sections of Chapter 11. Unit quaternions, that is, members of

, were called

*versors* by their discoverer, William Rowan Hamilton [3]. Further explanation can also be found in [4].

**Step 2: Understanding Triangle Congruences**Now click the "show congruence" checkbox to see triangles

,

, and

. The spherical triangle relates to ordinary three-dimensional rotations by taking heed that a rotation of angle

about an axis can be decomposed as the product of two rotations, each through a half-turn, about axes through the end points of any arc section of length

of a great circle in the plane defined by the axis on the unit sphere. Thus a rotation through an angle of twice the arclength of

about the axis defined by

can be realized by a rotation through

radians about the axis

followed by a second rotation through

radians about the axis

. Slide the "time" slider to see the triangle

turn red and rotate about

onto the central triangle, then rotate about

onto

; triangle

then turns green and rotates along the arc

** **to triangle

to show the effect of the composition of the two rotations by

radians. The triangles

,

, and

are all congruent to the unit quaternion multiplication triangle with vertices

.

,

, and

are the images of triangle

* *after the latter has been rotated

radians about the axis defined by the unit vectors

,

, and

, respectively. Penrose [2] describes them as the central triangle

having been "reflected" in vertices

,

, and

, respectively. So, the composition of a rotation through

about axis

followed by a

radian rotation about

maps triangle

onto

. The product of two rotations is again a rotation, and any rotation of the unit sphere is uniquely defined by a spherical triangle and its image under the rotation. Therefore,

is the triangle

after the latter has been rotated about the axis normal to the great circle containing the arc

, but the magnitude of the rotation, given

, by inspection is twice the arclength of the arc

. The unit quaternion

thus represents an

rotation about its axis, but through an angle that is twice the arclength of the quaternion itself. This is the reason why an element of

is often written with an angle of

; the angle

is the angle of the

rotation represented by the quaternion. Likewise

and

are the images of

and

under rotations about axes defined by the quaternions

and

, respectively. And, of course, the rotation mapping

onto

is the composition of the two rotations mapping first

to

, then

to

. So this rotation, represented by the quaternion

, can be found graphically as the great circle arc closing the spherical triangle

when we pay proper heed to the scaling factor

between the unit quaternion's arclength and the rotation's angle.

Further explanation is to be found at [4].

[1] C. Doran and A. Lasenby, Chap. 1, esp. §11.4 in

*Geometric Algebra for Physicists*, Cambridge: Cambridge University Press, 2003.

[2] R. Penrose, Chap. 11 in

*The Road to Reality: A Complete Guide to the Laws of the Universe*, London: Jonathan Cape, 2004.

[3] W. R. Hamilton, Chap. 1, §§8, 9 in

*Elements of Quaternions* (W. E. Hamilton, ed.), London: Longmans, Green & Co., 1866 pp. 133–157.