What project map creates maps the most distorted near the poles

One way of classifying map projections is by the type of the developable surface onto which the reference sphere is projected. A developable surface is a geometric shape that can be laid out into a flat surface without stretching or tearing. The three types of developable surfaces are cylinder, cone and plane, and their corresponding projections are called cylindrical, conical and planar. Projections can be further categorized based on their point(s) of contact (tangent or secant) with the reference surface of the Earth and their orientation (aspect).

Keep in mind that while some projections use a geometric process, in reality most projections use mathematical equations to transform the coordinates from a globe to a flat surface. The resulting map plane in most instances can be rolled around the globe in the form of cylinder, cone or placed to the side of the globe in the case of the plane. The developable surface serves as a good illustrative analogy of the process of flattening out a spherical object onto a plane.