Crooked Church Spire

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A crooked (or twisted) spire is a pyramidal tower with a twist relative to its base. This can be by design or as a result of changes over time caused by weathering [1].


This Demonstration simulates the twisting process of a pyramidal tower of fixed height caused by an expansion of its lateral edges due to faulty materials (undried wood), temperature, or humidity.

A classic example of a crooked church tower is the one in Chesterfield, Derbyshire, England [2].


Contributed by: Erik Mahieu (June 2015)
Open content licensed under CC BY-NC-SA



The twisted lateral edges of a right pyramid have the parametric equation of a conical spiral:


The base of the pyramid is a -sided regular polygon (and is also is the number of lateral faces of the pyramid), is the height of the apex, and (equal to in this Demonstration) is the radius of the circle through the vertices of the base.

Using the built-in Mathematica function ArcLength, the length of the lateral edges can be calculated using the formula:


From this, one can calculate a numerical approximation of the number of twists over as a function of the arc length . Using regression analysis, one finds this fitted model:


The twist angle is taken to be or the angle by which the pyramid is twisted relative to its base.


[1] Friends of Chesterfield Parish Church. "Notes on the Church." (Jun 9, 2015)

[2] Wikipedia. "Church of St Mary and All Saints, Chesterfield." (Jun 9, 2015) _Mary _and _All _Saints,_Chesterfield.

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