Pensieve header: The Schwarz Lantern, following http://blogs.scientificamerican.com/roots-of-unity/2013/11/30/counterexamples-in-origami/.

SchwarzLantern[n_, H_, ϵ_] := Module[{},
θ=π/n; Ts={};
For[p=0, p<2n H/ϵ, ++p,
h=ϵ p/n;
For[q=0, q<n, ++q,
s=(2q+p)θ;
AppendTo[Ts, Polygon[{
{Cos[s], Sin[s], h},
{Cos[s+θ], Sin[s+θ], h+ϵ/n},
{Cos[s+2θ], Sin[s+2θ], h}
}]];
AppendTo[Ts, Polygon[{
{Cos[s+θ], Sin[s+θ], h+ϵ/n},
{Cos[s+2θ], Sin[s+2θ], h},
{Cos[s+3θ], Sin[s+3θ], h+ϵ/n}
}]];
]
];
Ts
]

Table[Graphics3D[SchwarzLantern[5,1,ϵ]], {ϵ, {0.1,0.2,0.5,1,2}}]

ϵ=1;
Column[{
Slider[Dynamic[ϵ]],
Dynamic[Graphics3D[SchwarzLantern[5,1,ϵ]]]
}]