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1. What is the maximum $m$ such that the simplex with $n$ vertex points of form $$[00\dots00],[10\dots00],\dots,[11\dots1100\dots00],\dots,[11\dots11]\in\{0,1\}^{n-1}$$ have a non-singular linear transformation whose projection yields boundary of a regular $m$-gon on $2D$ plane?

2. Without the vertex restrictions is it always possible to have a simplex in $n$ dimensions whose projection is a regular $n$-gon?

If 2. is ok then perhaps only a linear transformation might need to be searched which is quite straightforward. $m\geq n-1$ would hold in that case.

What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?

1. What is the maximum $m$ such that the simplex with $n$ vertex points of form $$[00\dots00],[10\dots00],\dots,[11\dots1100\dots00],\dots,[11\dots11]\in\{0,1\}^{n-1}$$ have a non-singular linear transformation whose projection yields boundary of a regular $m$-gon on $2D$ plane?

2. Without the vertex restrictions is it always possible to have a simplex in $n$ dimensions whose projection is a regular $n$-gon?

If 2. is ok then perhaps only a linear transformation might need to be searched. $m\geq n-1$ would hold in that case.

1. What is the maximum $m$ such that the simplex with $n$ vertex points of form $$[00\dots00],[10\dots00],\dots,[11\dots1100\dots00],\dots,[11\dots11]\in\{0,1\}^{n-1}$$ have a non-singular linear transformation whose projection yields boundary of a regular $m$-gon on $2D$ plane?

2. Without the vertex restrictions is it always possible to have a simplex in $n$ dimensions whose projection is a regular $n$-gon?

If 2. is ok then perhaps only a linear transformation might need to be searched which is quite straightforward. $m\geq n-1$ would hold in that case.

What is the maximum $m$ such that the simplex with $n$ vertex points of form $$[00\dots00],[10\dots00],\dots,[11\dots1100\dots00],\dots,[11\dots11]\in\{0,1\}^{n-1}$$ have a non-singular linear transformation whose projection yields boundary of a regular $m$-gon on $2D$ plane?

1. What is the maximum $m$ such that the simplex with $n$ vertex points of form $$[00\dots00],[10\dots00],\dots,[11\dots1100\dots00],\dots,[11\dots11]\in\{0,1\}^{n-1}$$ have a non-singular linear transformation whose projection yields boundary of a regular $m$-gon on $2D$ plane?

2. Without the vertex restrictions is it always possible to have a simplex in $n$ dimensions whose projection is a regular $n$-gon?

If 2. is ok then perhaps only a linear transformation might need to be searched. $m\geq n-1$ would hold in that case.