Not to repeat halcyon, but using increasingly more points actually adds to the dimension of the figure:

Two points makes a line segment.

Three points makes a triangle with triangular gaps (see w/u by just a guy above).

Four points makes a tetrahedron with octehedral gaps (an octehedron bounded by a triangle, the shape 'removed' from the center of the tetrahedron below):

```                                       ,|,
,7`\,\,
,7`  `| `\,
,7`     \,  `\,
,7`       `|    `\,
,7`          \,     `\,
,7`            `|       `\,
,7`               \,        `\,
,7`                 `|        /7`\,
,7`                    \,      AV `|`\,
,7`'TTs.,                `|     /7   \, `\,
,7` \\,  `'TTs.,            \,   AV    `|   `\,
,7`   `|`\,     `'TTs.,       `|  /7      \,    `\,
,7`      \, `\,        `'TTs.,   \,AV       `|      `\,
,7`        `|   `\,        ,7``'TTs`|7         \,       `\,
,7`           \,    `\,    ,7`      ,7\\,        `|         K`
,7`             `|      `\,,7`      ,7` `|`\,       \,       AV
,7`                \,       AV'TTs.,,7`    \, `\,     `|      /7
,7`                  `|      /7     ,7`      `|   `\,    \,    AV
,T,                    \,    AV    ,7`         \,    `\,  `|   /7
`'TTs.,                `|   /7   ,7`           `|      `\, \, AV
`'TTs.,            \, AV  ,7`              \,       `\||/7
`'TTs.,       `|/7 ,7`                `|         `AV
`'TTs.,   \,,7`                   \,        /7
`'TTs`|                      `|       AV
`'TTs.,                  \,     /7
`'TTs.,             `|    AV
`'TTV.,         \,  /7
`'TTs.,    `| AV
`'TTs.,\/7
`'T`

The shape from the middle:

____,,..----**/7,
____,,..----**'`             AV`|
|'TTs.,                       /7  \,
\,   `'TTs.,                 AV   `|
`|        `'TTs.,           /7     \,
\,            `'TTs.,     AV      `|
`|                 `'TTs./7        \,
\,                    ,7\,        `|
`|                  ,7`  `\,       \,
\,               ,7`      `\,     `|
`|             ,7`          `\,    \,
\,          ,7`              `\,  `|
`|        ,7`                  `\, \,
\,     ,7`                      `\||
`|   ,7`              ____,,..----**
\,,7`____,,..----**'`
7*'`

```

(In the above figure, each face of the tetrahedron is a Sierpinski triangle, and every 3D gap is octehedral in shape.) It is clear from the upper shape above that four generating points could easily fill the quadrilateral formed by them if the points are randomly placed (percentage chances? I have no idea...).

Etc. A three dimensional screen would be more suited for viewing this fractal with four generating points, but the 2-D version is often recognizeable. Also, it often helps to have a color change when approaching any one of the points.