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A tapping head is a drill or milling machine attachment used to cut threads of a nut. Instead of being a solid surface on the inside like the type of nuts used with Phillips or flat head screws, there are threads drilled through the inside. The three most common types of tapping heads are bottom tapping or plug tap, intermediate tap or second tap, and taper tap.
Bottom tapping or plug taps are used to form tapping heads that have threads with little to no taper. This allows the screw to be successfully installed and is usually used for threading into a blind hole. A blind hole is one that has been drilled to a certain depth but not all the way through the wood or other material. Bottom tapping is also sometimes used in conjunction with other tapping types because the bottom tapping end is short and may not allow for adequate threading.
Intermediate or second tapping refers to a tapered tap that is smaller in diameter than the bottom tap. Named a second tap since it is often used in conjunction with other taps during the threading process, this tap typically creates the hole later used by a bottom tapping or taper tap. Intermediate or second taps are the most common in the US.
A taper tap is much smaller than its counterparts. Used on harder surfaces such as steel and other types of metal, the taper tap has a sharp, pointed end and small diameter. A taper tap is also used when the area is hard to reach or the thread count required is higher, since most taper taps create as much as 10 threads.
The tapping heads formed by these taps include procunier tapping head, tapmatic tapping head, hex head tapping screw, flat head tapping screw and reversing tapping head. A procunier tapping head is named after William Procunier, who manufactured positive drive reversing tapping heads in the 1920s. Tapmatic tapping heads are manufactured by the Tapmatic company and feature self-reversing tapping heads. Hex head tapping screws have a hexagon-shaped slot in the top where the screwing tool is inserted to tighten the screw in place.
A flat head tapping screw head is similar to a traditional screw with a single slot to tighten. The main difference is the tapping screw is designed to drill into a previously started hole. A traditional screw works equally well whether or not the hole is predrilled. A reversing tapping head, as the name suggests, is a tapping head that can be drilled in or out at the same speed without the need for stopping and restarting the drill.
What you are describing as a tapping head is simply called a tap weather bottom or starter. A tapping head is an attachment you put the tap into and it drives the tap into the hole you want tapped as opposed to tapping it manually.