4 just some minor reorganization/clarification
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for simplicity, assume you share your mount's space during combat

Yes, the line you quoteThis means that you fillyour Space stat takes on the entire spacevalue of your mount’s Space stat, i.e. 15 ft. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them.

\begin{align} { \text{Without reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & { \text{With reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{align} Where So if \$•\$ is where you are and \$○\$ is a square you threaten.,

\begin{array}{c c|c c} { \text{Without reach weapon:} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & \quad& { \text{With reach weapon:} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{array}

Creatures do not need reach to strike you; in fact, these same diagrams work for them. If they stand on a \$○\$ then they can reach one of the \$•\$ squares, i.e. they can reach you (use the grid that corresponds to whether or not they have a reach weapon; your weapon is irrelevant to their reach). However, since your space is larger, they can actually stand closer to you with a reach weapon and still strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

But that Pathfinder implements an abstraction does not have to be the end of the story. As HeyICanChan points out, Paizo creative director James Jacobs does not consider the abstractions around mounted combat in general to truly be sufficient, suggesting that

Mounted combat is underdetailed in Pathfinder. So, the more you get into it, the more you're going to have to house rule.

In short, Paizo chose simpler, quicker-to-learn, quicker-to-run abstractions for mounted combat, quite possibly because they were focused on more typical cases and didn’t want to bog the system down with special rules for scenarios that may very well not come up in the majority of games (since most riders are just one size category smaller than their mount and therefore the abstraction doesn’t really look too weird). You can choose to do differently, especially if your game isn’t going to look like “most” Pathfinder games and smaller creatures riding very big ones is going to be an important part of it.

But be aware that abstractions are powerful and useful, and reducing abstraction thresholds means greater costs in time to learn, time to run, and complexity. Paizo didn’t think those costs were worth it, at least for their idea of “most” Pathfinder games. You might think they are worth it, at least for your game. But don’t ignore the fact that, worth it or not, the costs are still there. You need to think about them while designing any houserule you want to implement.

Yes, the line you quote means that you fill the entire space. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them.

\begin{align} { \text{Without reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & { \text{With reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{align} Where \$•\$ is where you are and \$○\$ is a square you threaten.

Creatures do not need reach to strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

But that Pathfinder implements an abstraction does not have to be the end of the story. As HeyICanChan points out, Paizo creative director James Jacobs does not consider the abstractions around mounted combat in general to truly be sufficient, suggesting that

Mounted combat is underdetailed in Pathfinder. So, the more you get into it, the more you're going to have to house rule.

In short, Paizo chose simpler, quicker-to-learn, quicker-to-run abstractions for mounted combat, quite possibly because they were focused on more typical cases and didn’t want to bog the system down with special rules for scenarios that may very well not come up in the majority of games (since most riders are just one size category smaller than their mount and therefore the abstraction doesn’t really look too weird). You can choose to do differently, especially if your game isn’t going to look like “most” Pathfinder games and smaller creatures riding very big ones is going to be an important part of it.

But be aware that abstractions are powerful and useful, and reducing abstraction thresholds means greater costs in time to learn, time to run, and complexity. Paizo didn’t think those costs were worth it, at least for their idea of “most” Pathfinder games. You might think they are worth it, at least for your game. But don’t ignore the fact that, worth it or not, the costs are still there. You need to think about them while designing any houserule you want to implement.

for simplicity, assume you share your mount's space during combat

This means that your Space stat takes on the value of your mount’s Space stat, i.e. 15 ft. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them. So if \$•\$ is where you are and \$○\$ is a square you threaten,

\begin{array}{c c|c c} { \text{Without reach weapon:} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & \quad& { \text{With reach weapon:} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{array}

Creatures do not need reach to strike you; in fact, these same diagrams work for them. If they stand on a \$○\$ then they can reach one of the \$•\$ squares, i.e. they can reach you (use the grid that corresponds to whether or not they have a reach weapon; your weapon is irrelevant to their reach). However, since your space is larger, they can actually stand closer to you with a reach weapon and still strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

But that Pathfinder implements an abstraction does not have to be the end of the story. As HeyICanChan points out, Paizo creative director James Jacobs does not consider the abstractions around mounted combat in general to truly be sufficient, suggesting that

Mounted combat is underdetailed in Pathfinder. So, the more you get into it, the more you're going to have to house rule.

In short, Paizo chose simpler, quicker-to-learn, quicker-to-run abstractions for mounted combat, quite possibly because they were focused on more typical cases and didn’t want to bog the system down with special rules for scenarios that may very well not come up in the majority of games (since most riders are just one size category smaller than their mount and therefore the abstraction doesn’t really look too weird). You can choose to do differently, especially if your game isn’t going to look like “most” Pathfinder games and smaller creatures riding very big ones is going to be an important part of it.

But be aware that abstractions are powerful and useful, and reducing abstraction thresholds means greater costs in time to learn, time to run, and complexity. Paizo didn’t think those costs were worth it, at least for their idea of “most” Pathfinder games. You might think they are worth it, at least for your game. But don’t ignore the fact that, worth it or not, the costs are still there. You need to think about them while designing any houserule you want to implement.

3 addressing comments about houserules
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Yes, the line you quote means that you fill the entire space. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them.

\begin{align} { \text{Without reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & { \text{With reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{align} Where \$•\$ is where you are and \$○\$ is a square you threaten.

Creatures do not need reach to strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

But that Pathfinder implements an abstraction does not have to be the end of the story. As HeyICanChan points out, Paizo creative director James Jacobs does not consider the abstractions around mounted combat in general to truly be sufficient, suggesting that

Mounted combat is underdetailed in Pathfinder. So, the more you get into it, the more you're going to have to house rule.

In short, Paizo chose simpler, quicker-to-learn, quicker-to-run abstractions for mounted combat, quite possibly because they were focused on more typical cases and didn’t want to bog the system down with special rules for scenarios that may very well not come up in the majority of games (since most riders are just one size category smaller than their mount and therefore the abstraction doesn’t really look too weird). You can choose to do differently, especially if your game isn’t going to look like “most” Pathfinder games and smaller creatures riding very big ones is going to be an important part of it.

But be aware that abstractions are powerful and useful, and reducing abstraction thresholds means greater costs in time to learn, time to run, and complexity. Paizo didn’t think those costs were worth it, at least for their idea of “most” Pathfinder games. You might think they are worth it, at least for your game. But don’t ignore the fact that, worth it or not, the costs are still there. You need to think about them while designing any houserule you want to implement.

Yes, the line you quote means that you fill the entire space. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them.

\begin{align} { \text{Without reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & { \text{With reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{align} Where \$•\$ is where you are and \$○\$ is a square you threaten.

Creatures do not need reach to strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

Yes, the line you quote means that you fill the entire space. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them.

\begin{align} { \text{Without reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & { \text{With reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{align} Where \$•\$ is where you are and \$○\$ is a square you threaten.

Creatures do not need reach to strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

But that Pathfinder implements an abstraction does not have to be the end of the story. As HeyICanChan points out, Paizo creative director James Jacobs does not consider the abstractions around mounted combat in general to truly be sufficient, suggesting that

Mounted combat is underdetailed in Pathfinder. So, the more you get into it, the more you're going to have to house rule.

In short, Paizo chose simpler, quicker-to-learn, quicker-to-run abstractions for mounted combat, quite possibly because they were focused on more typical cases and didn’t want to bog the system down with special rules for scenarios that may very well not come up in the majority of games (since most riders are just one size category smaller than their mount and therefore the abstraction doesn’t really look too weird). You can choose to do differently, especially if your game isn’t going to look like “most” Pathfinder games and smaller creatures riding very big ones is going to be an important part of it.

But be aware that abstractions are powerful and useful, and reducing abstraction thresholds means greater costs in time to learn, time to run, and complexity. Paizo didn’t think those costs were worth it, at least for their idea of “most” Pathfinder games. You might think they are worth it, at least for your game. But don’t ignore the fact that, worth it or not, the costs are still there. You need to think about them while designing any houserule you want to implement.

2 adding diagrams
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Yes, the line you quote means that you fill the entire space. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them.

\begin{align} { \text{Without reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & { \text{With reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{align} Where \$•\$ is where you are and \$○\$ is a square you threaten.

Creatures do not need reach to strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

Yes, the line you quote means that you fill the entire space. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them.

Creatures do not need reach to strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

Yes, the line you quote means that you fill the entire space. Your reach doesn’t change, so in this case you have a 15-ft. space with a 5-ft. reach, so you threaten the 16 squares surrounding the 3×3 square. If you have a reach weapon, you don’t threaten those 16 squares, but do threaten the 20 squares surrounding them.

\begin{align} { \text{Without reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & • & • & • & ○ \\ \hline ○ & ○ & ○ & ○ & ○ \\ \end{array} \\ } & \quad & { \text{With reach} \\ \begin{array}{c|c} ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & • & • & • & & ○ \\ \hline ○ & & & & & & ○ \\ \hline ○ & ○ & ○ & ○ & ○ & ○ & ○ \\ \end{array} } \end{align} Where \$•\$ is where you are and \$○\$ is a square you threaten.

Creatures do not need reach to strike you.

This is all kind of weird and very abstract, but Pathfinder is a pretty abstract game in a lot of ways. This simplification is really just an expansion of the already-quite-abstract grid system where a typical person is said to “take up” a 5-ft. cube (I doubt anyone in the history of Homo sapiens has ever actually been 5 feet broad or wide). Basically, you take up that space because you are constantly moving and this is the area you “control” and can therefore move freely in (read: not require an action, not provoke an attack of opportunity). With a mount, which controls its own space, you are able to move around freely atop the mount and so take advantage of the control over that space that it provides.

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