How to construct Gothic arches
A Gothic arch is a sharp-pointed arch, formed of two arc segments (parts of a circle). The lower part of the arch is parallel sided, up to the level of the springing points.
The Gothic arch evolved from the round-topped Roman arch. It has two advantages; one is obvious, in that a Gothic arch reaches higher for a given width. Another advantage is less visible. A major consideration when building a masonry arch is the amount of horizontal thrust that it produces on its foundations. Masonry walls can easily absorb large vertical compressive forces, but a sideways force will tend to topple the wall, unless braced by external buttresses. A Gothic arch may produce only half the side-thrust of a similar Roman arch.
Medieval masons worked with few tools, and laid out almost all their work with compasses. The arch is no exception. They also had a great fondness for certain geometric constructions, and took great interest (particularly in later centuries) in relating these back to Greek mathematics and philosophy.
Gothic archs are laid out as two simple arcs. To give a smooth curve at the base of the arch, the centre for these arcs must obviously be on the same level as the springing points. Varying the arc length (and the position of the centre) relative to the height of the arch will control the proportions and height of the arch.
The classic Gothic arch is known as the quinto acuto or "pointed fifth". The arcs are 4/5 of the span. The centres of the arcs are inside the span of the arch.
A lower style of arch is the mezzo acuto. This is halfway between a semicircular arch anf the pointed fifth.
In the 14th century, the construction of the Duomo in Florence ran into yet more constructional problems when the architect Filippo Brunelleschi was acused of building the dome "half round" instead of the agreed quinto acuto.
Another classic arch is the recto. Its characteristic is that its height (above the springing points) is equal to its width. This arch was more commonly seen in windows, particularly where narrow recto arches were used as tracery within an overall frame of quinto acuto.
To construct a recto arch, the arc length should be 5/4 of the width of the arch. The arc centres are outisde the span.
Another feature of the recto arch is that the apex of the arch and the arc centre describe a 3:4:5 triangle (shown in red on the diagram), the right-agled triangle well known in Masonic arcana.
The tallest common Gothic arch uses an arc radius equal to twice the span. This gives a height of 1.1 spans (square root (5) / 2).
A common feature of Gothic arched windows is for them to have internal tracery made of small ribbed arches. These arches are themselves Gothic in form, but usually of narrower proportion than the outside arch and with the springing points at the same height as those of the window.