Both methods turn a flat filter paper disc into a conical shape fitting a standard 60° funnel so it can hold more solution and use gravity filtration more efficiently.
While a quadrant folded filter paper is a disk simply pre-folded in a quarter of a circle, a fluted filter paper has been pre-pleated to minimize the contact surface of the filter with the funnel, decreasing filtration time.
Quadrant Folded Paper
There are a few variants of the quadrant or quarter folded filter paper method, all turns a disk into a smooth conical shape that fir a standard funnel. Folding is fairly easy but it takes some extra steps to make sure the conical shape remains in place while pouring the solution.
Fluted Filter Paper
Fluted filter paper is a disc that has been pre-pleated, forming a sort of accordion to maximize the effective filtration surface of the filter. Even though there are a few methods to fold a disk in 8 pleats, it's not an easy task and it takes some time. Most fluted filter paper sold by brands uses machines to ensure a higher consistency of the shape and a more accurate position and size of the pleats. Small discs (<100 mm diameter) often have 8 pleats, larger diameters can have from 10 to 16 pleats, maintaining a well defined conical shape. This prevent the filter from sticking to the funnel, increasing the surface area that can effectively filter the solution thus speeding up filtration.
Gravity Filtration Only
Fluting of folding a disk increase gravity filtration efficiency compared to a flat disk, but it can't be used for vacuum (suction) filtration which requires a flat Büchner funnel.
