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Here is a 100px × 100px div with a 30px border:

height: 100px;
width: 100px;
border: 30px solid #999;

Watch what happens when each border has a different colour:

height: 100px;
width: 100px;
border-top: 30px solid #C66;
border-bottom: 30px solid #6C6;
border-left: 30px solid #66C;
border-right: 30px solid #CC6;

Now if we drop the setting of height and explicitly set the width of the div to be 0, we get the following:

width: 0;
border-top: 30px solid #C66;
border-bottom: 30px solid #6C6;
border-left: 30px solid #66C;
border-right: 30px solid #CC6;

Drop the top border and make the left and right borders transparent and you get this:

width: 0;
border-bottom: 30px solid #6C6;
border-left: 30px solid transparent;
border-right: 30px solid transparent;

The side borders don’t have to be the same size as the bottom border. Here’s 30px bottom with 52px sides:

width: 0;
border-bottom: 30px solid #6C6;
border-left: 52px solid transparent;
border-right: 52px solid transparent;

And here’s a div with a top border instead of a bottom border:

width: 0;
border-top: 30px solid #6C6;
border-left: 52px solid transparent;
border-right: 52px solid transparent;

Put a 104px × 60px div with a background colour between them and you get:

width: 0;
border-bottom: 30px solid #6C6;
border-left: 52px solid transparent;
border-right: 52px solid transparent;

width: 104px;
height: 60px;
background-color: #6C6;

width: 0;
border-top: 30px solid #6C6;
border-left: 52px solid transparent;
border-right: 52px solid transparent;

And that’s how we get a hexagon in CSS. The 30:52 ratio in the border widths is approximately 1:√3 which is ratio required for a hexagon.

A similar approach can be used to get a hexagon rotated 30°. We just flip around some of the directions, use float: left and drop the explict setting of width to 0.

float: left;
border-right: 30px solid #6C6;
border-top: 52px solid transparent;
border-bottom: 52px solid transparent;

float: left;
width: 60px;
height: 104px;
background-color: #6C6;

float: left;
border-left: 30px solid #6C6;
border-top: 52px solid transparent;
border-bottom: 52px solid transparent;

Both orientations of hexagons can easily be tiled. The first orientation involves a margin-bottom of -26px and margin-left of 3pxon each hexagon and a margin-left of 53px on even rows:

.hex {
    float: left;
    margin-left: 3px;
    margin-bottom: -26px;
}
.hex .top {
    width: 0;
    border-bottom: 30px solid #6C6;
    border-left: 52px solid transparent;
    border-right: 52px solid transparent;
}
.hex .middle {
    width: 104px;
    height: 60px;
    background: #6C6;
}
.hex .bottom {
    width: 0;
    border-top: 30px solid #6C6;
    border-left: 52px solid transparent;
    border-right: 52px solid transparent;
}
.hex-row {
    clear: left;
}
.hex-row.even {
    margin-left: 53px;
}

The second orientation involves a margin-right of -26px and margin-bottom of -50px on each hexagon as well as a margin-top: 53px on even columns:

.hex {
    float: left;
    margin-right: -26px;
    margin-bottom: -50px;
}
.hex .left {
    float: left;
    width: 0;
    border-right: 30px solid #6C6;
    border-top: 52px solid transparent;
    border-bottom: 52px solid transparent;
}
.hex .middle {
    float: left;
    width: 60px;
    height: 104px;
    background: #6C6;
}
.hex .right {
    float: left;
    width: 0;
    border-left: 30px solid #6C6;
    border-top: 52px solid transparent;
    border-bottom: 52px solid transparent;
}
.hex-row {
    clear: left;
}
.hex.even {
    margin-top: 53px;
}

And to finish things off, here’s a quick demo of a CSS 3D perspective transform applied to the hex grid:

-webkit-transform: perspective(600px) rotateX(60deg);
-moz-transform: perspective(600px) rotateX(60deg);
-ms-transform: perspective(600px) rotateX(60deg);
-o-transform: perspective(600px) rotateX(60deg);
transform: perspective(600px) rotateX(60deg);

Addendum

Will Hardy suggested on Twitter the following use of :before and :after to reduce the necessary divs to one:

.hex:before {
    content: " ";
    width: 0; height: 0;
    border-bottom: 30px solid #6C6;
    border-left: 52px solid transparent;
    border-right: 52px solid transparent;
    position: absolute;
    top: -30px;
}

.hex {
    margin-top: 30px;
    width: 104px;
    height: 60px;
    background-color: #6C6;
    position: relative;
}

.hex:after {
    content: "";
    width: 0;
    position: absolute;
    bottom: -30px;
    border-top: 30px solid #6C6;
    border-left: 52px solid transparent;
    border-right: 52px solid transparent;
}

jawns pointed out on Hacker News that there's a Unicode code point U+2B22: