Updated date:

What Is a Pendentive Form and How to 3d-Model It

I work as seminar lecturer and literary translator. I am a published author, and graduate of the University of Essex (BA in Philosophy).

A very Byzantine design

The pendentive is a highly elegant architectural formation, and one which also has notable practical value. It was first used in the Byzantine Empire, and the most famous example of a structure with pendentives is the Hagia Sophia cathedral, in Constantinople. The pendentive is what allows for the massive dome to be placed in a way that it doesn’t require high supporting walls which would limit the view for those being inside the building, and the consequent lack of obvious means for support gives the impression that the vast dome is hovering in mid-air. In older dome architecture – for example in the Pantheon in Rome – less elegant solutions were used; the Pantheon had to surround its dome with a wall, and other kinds of domed buildings would place the dome on a square (or other polygonal) base, thus limiting its visibility to anyone who would be inside the building.

Inside-out drawing of Hagia Sophia, by Stephen Biesty

Inside-out drawing of Hagia Sophia, by Stephen Biesty

Step 1 in modeling a pendentive

The pendentive is based on three forms: a large sphere, a cube inscribed to that sphere, and a cylinder which is inscribed to that cube. An inscribed form is a form which is perfectly contained within another, while also being the largest possible form to be thus contained: in other words, the cube has all its vertices touching the periphery of the sphere it is inscribed on, and the same is the case regarding the cylinder inscribed to the square. This is presented in Figure A.

Figure A: Sphere, cube inscribed to the sphere, cylinder inscribed to the cube

Figure A: Sphere, cube inscribed to the sphere, cylinder inscribed to the cube

Step 2 in modeling a pendentive

The pendentive will be constructed by carving the volume of the large sphere. This is to be done by taking out cylindrical volumes, of cylinders which are in all other ways identical to the one we inscribed to the cube, other than being extended a bit in regards to height. In total there are three such cylinders to subtract from the sphere:

-Two from the sides and inwards

-One from the center and upwards

The correct placement of those three cylinders, prior to subtracting them from the sphere, is shown in Figure B. Notice that the original cylinder was extended in height in order to allow for enough volume to go beyond the sphere we want to carve (otherwise we would be only taking parts out of its inside, and the result wouldn’t open up any space in the circumference of the sphere.

Figure B: preparing to substract the cylinders from the volume of the sphere

Figure B: preparing to substract the cylinders from the volume of the sphere

Step 3 in modeling a pendentive

At this point we can erase the cube, because we only needed it so as to calculate the position of the cylinder.

Figure C presents the result, when the three cylinders have been subtracted from the volume of the sphere. In Blender this process is done by using the option for “Boolean operations”. In all 3d modeling programs the method rests on Boolean logic, and basic subtraction, addition or intersection between the volumes acted upon.

Figure C: the cylinders have been subtracted

Figure C: the cylinders have been subtracted

Final step in modeling a pendentive

Now all that remains is for us to erase the bottom (symmetrical) part of our new volume. Figure D presents the final shape.

Our pendentive is ready for use!

Figure D: The pendentive has been modelled!

Figure D: The pendentive has been modelled!

An article on modeling other Byzantine architectural forms

© 2018 Kyriakos Chalkopoulos