
Gauss |

What your player may be recalling is that you count as if your space is the same as the mounts space. This does not increase your reach but does increase the area you can cover (assuming your mount is larger than yourself).
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Example: a Medium creature using a longsword can affect 9 squares.
XXX
XMX
XXX
A Medium creature using a longsword mounted on a Horse (large) can affect 16 squares.
XXXX
XHHX
XHHX
XXXX

Serisan |

Thinking 3D doesn't necessarily make sense a lot of the time. Perfect example: I have a 6' tall Half-Orc Medium (the Occult playtest class, not the size). At 7th level, using The Bear and seancing it for the level 13 ability, I become Huge, which puts me up to 24' tall (round to 25'). You'd think my natural reach at this point would be an additional 15' up from there, but I technically don't occupy 10' of height and can only reach 6' above my head.
That said, a medium-sized character on a large-sized mount should threaten based on being in any of the 8 cubes that the large mount occupies, which actually extends the reach out a bit further since that puts a standard melee weapon (non-reach) at the 15' increment from the ground. To put it in topographical terms, with M = Mount and T = Threaten...
TTTT 0-5' above ground level
TMMT
TMMT
TTTT
TTTT 5-10' above ground level
TMMT
TMMT
TTTT
TTTT 10-15' above ground level
TTTT
TTTT
TTTT

Gauss |

Serisan, thinking 3d makes sense within the (necessarily abstract) game rules. Once you go into the details of height, weight, girth, length, etc. things will break down for you. It is best not to go there.
Yes, a medium sized character with a reach of 5' on a large size mount affects 4 levels worth of squares (0 to -5, 0 to 5, 5 to 10, and 10 to 15).

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Indeed.
Think in 5 foot cubes. Like Minecraft.
Seriously. It'll make things so much easier to visualize.
Your 6 foot Half-orc is really 5 feet wide, and five feet tall.
Even when laying prone.
He can even reach anywhere into any adjacent 5 foot cube.
It'll blow your mind at first, but you get used to it.