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(I have yet to see an antenna like that -- it may exist only in math theory.)
A higher dBi is obtained by designing the antenna so that the strength of the signal is more focused.
The little vertical 'rubber duck' antenna that you see on most wireless routers or Access Points (AP's) focusses the signal such that it is stronger from the sides of the antenna and weaker above and below. It is omni-directional. If the signals were visible, it would resemble a fat donut shape with the radiator in the middle. This focusing gives a relative strength of 2.15 dBi.
After-market vertical antennas get stronger dBi by narrowing the beam. The signals can be "heard" more from farther away, but less from above and below. If the signals were visible, it would look like a bigger donut, but with skinnier dough.
Directional antennas are designed to focus the beam such that it prefers one direction from the antenna. If the signals were visible, it would look like a bulb syringe with the radiator at the narrow-skinny end.
Choosing between directional or omni-directional usually has a lot to do with where the device is going to be in your home or office. If the wireless router or AP is in a corner near an outside wall, a directional antenna pointed in toward the bulk of your space is usually a better choice. But if it is near the center of the space, then omni-directional is probably the right answer.
editorial comment: This link adds some pictures and colors to funchords ' very nice FAQ. »antennas.ee.duth.gr/TheLaborator···hort.htm
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