Article of the month January

I have always wondered why, on a declining vertical sundial, the XII-hours line is still vertical. I expected the whole hour line pattern to shift - including that line.
How would one explain, in one paragraph, why it does not?

The plane that contains the pole style is the north-south plane. When the sun souths, the pole style shadow is also in the north-south plane. That plane intersects a vertical wall - of whatever declination - along a vertical line through the point where the pole style is fixed to the wall.

Why did I expect otherwise? Probably because for practical purposes, the (dominantly present) style triangle is fitted at right angles to the sundial face, and therefore itself not in the north-south plane.

The following drawing, in which H is a horizontal plane and V a vertical declining sundial face, explains things.

AB = pole style
AD = substyle
AC = XII-hours line
ABC = vertical north-south plane
ABD = style triangle, perpendicular to the sundial face

Hans de Rijk