The usual projection of the Fano plane. Hover over a node to see the nodes on the line dual to that node.

fano_plane.svg

<?xml version="1.0" encoding="UTF-8" ?>
<svg xmlns="http://www.w3.org/2000/svg" width="400" height="400" viewBox="0 0 200 200">

<title>Fano plane</title>
<!-- A diagram of the Fano finite projective plane using an equilateral triangle.
    Created by PM 2Ring 2017.10.02
-->

<!-- A solid white background -->
<!-- <rect width="200%" height="200%" fill="#fff"/> -->

<g transform="translate(100, 170)">

<!-- Draw the lines. 135.1 / 78 ~= sqrt(3) -->
<g style="stroke-width:1.5">
    <line x1="0" y1="-135.1" x2="78" y2="0" stroke="#000"/> <!--1-->
    <line x1="0" y1="-135.1" x2="-78" y2="0" stroke="#a00"/> <!--2-->
    <line x1="0" y1="-135.1" x2="0" y2="0" stroke="#f00"/> <!--3-->
    <line x1="-78" y1="0" x2="78" y2="0" stroke="#00a"/> <!--4-->
    <line x1="-39" y1="-67.55" x2="78" y2="0" stroke="#00f"/> <!--5-->
    <line x1="39" y1="-67.55" x2="-78" y2="0" stroke="#a0a"/> <!--6-->
    <circle cx="0" cy="-45.033" r="45.033" stroke="#f0f" fill="none"/> <!--7-->
</g> 

<!-- Draw the nodes -->
<g style="fill:#fff; font-family:helvetica; text-anchor:middle"
    font-size="15">
    <g>
        <circle cx="-78" cy="0" r="10" fill="#000"/>
        <text x="-78" y="5">1</text><title>246</title>
    </g>
    <g>
        <circle cx="78" cy="0" r="10" fill="#a00"/>
        <text x="78" y="5">2</text><title>145</title>
    </g>
    <g>
        <circle cx="0" cy="0" r="10" fill="#f00"/>
        <text x="0" y="5">3</text><title>347</title>
    </g>
    <g>
        <circle cx="0" cy="-135.1" r="10" fill="#00a"/>
        <text x="0" y="-130.1">4</text><title>123</title>
    </g>
    <g>
        <circle cx="-39" cy="-67.55" r="10" fill="#00f"/>
        <text x="-39" y="-62.55">5</text><title>257</title>
    </g>
    <g>
        <circle cx="39" cy="-67.55" r="10" fill="#a0a"/>
        <text x="39" y="-62.55">6</text><title>167</title>
    </g>
    <g>
        <circle cx="0" cy="-45.033" r="10" fill="#f0f"/>
        <text x="0" y="-40.033">7</text><title>356</title>
    </g> 
</g>
</g>
</svg>

The 3D version, PG(3,2):