Risto A. Paju<p>I've finally managed to put my entire "geodesic series" of himmelis side by side, thanks @noira_musti for the suggestion. The edge counts are 6, 12, 30, 36, 42, 48, 84, 90, 120, and 210.</p><p><a href="https://mathstodon.xyz/tags/himmeli" class="mention hashtag" rel="tag">#<span>himmeli</span></a> <a href="https://mathstodon.xyz/tags/puzuri" class="mention hashtag" rel="tag">#<span>puzuri</span></a> <a href="https://mathstodon.xyz/tags/strawart" class="mention hashtag" rel="tag">#<span>strawart</span></a> <a href="https://mathstodon.xyz/tags/geodesicseries" class="mention hashtag" rel="tag">#<span>geodesicseries</span></a> <a href="https://mathstodon.xyz/tags/geodesichimmeli" class="mention hashtag" rel="tag">#<span>geodesichimmeli</span></a> <a href="https://mathstodon.xyz/tags/geodesicpolyhedron" class="mention hashtag" rel="tag">#<span>geodesicpolyhedron</span></a> <a href="https://mathstodon.xyz/tags/polyhedron" class="mention hashtag" rel="tag">#<span>polyhedron</span></a> <a href="https://mathstodon.xyz/tags/geometricart" class="mention hashtag" rel="tag">#<span>geometricart</span></a> <a href="https://mathstodon.xyz/tags/algorithmicart" class="mention hashtag" rel="tag">#<span>algorithmicart</span></a> <a href="https://mathstodon.xyz/tags/algorist" class="mention hashtag" rel="tag">#<span>algorist</span></a> <a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#<span>mathart</span></a> <a href="https://mathstodon.xyz/tags/laskutaide" class="mention hashtag" rel="tag">#<span>laskutaide</span></a></p>
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#himmeli
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Risto A. Paju<p>No new himmelis here, but I thought it might be interesting to compare some of them side by side. In all of these himmelis, the long edges are 4.5 cm. The short edges are not much shorter, and they are in the minority anyway. The total edge counts are 30, 90, 120 and 210.</p><p><a href="https://mathstodon.xyz/tags/himmeli" class="mention hashtag" rel="tag">#<span>himmeli</span></a> <a href="https://mathstodon.xyz/tags/puzuri" class="mention hashtag" rel="tag">#<span>puzuri</span></a> <a href="https://mathstodon.xyz/tags/strawart" class="mention hashtag" rel="tag">#<span>strawart</span></a> <a href="https://mathstodon.xyz/tags/geodesichimmeli" class="mention hashtag" rel="tag">#<span>geodesichimmeli</span></a> <a href="https://mathstodon.xyz/tags/geodesicsphere" class="mention hashtag" rel="tag">#<span>geodesicsphere</span></a> <a href="https://mathstodon.xyz/tags/geodesicpolyhedron" class="mention hashtag" rel="tag">#<span>geodesicpolyhedron</span></a> <a href="https://mathstodon.xyz/tags/polyhedron" class="mention hashtag" rel="tag">#<span>polyhedron</span></a> <a href="https://mathstodon.xyz/tags/geometricart" class="mention hashtag" rel="tag">#<span>geometricart</span></a> <a href="https://mathstodon.xyz/tags/algorithmicart" class="mention hashtag" rel="tag">#<span>algorithmicart</span></a> <a href="https://mathstodon.xyz/tags/algorist" class="mention hashtag" rel="tag">#<span>algorist</span></a> <a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#<span>mathart</span></a> <a href="https://mathstodon.xyz/tags/laskutaide" class="mention hashtag" rel="tag">#<span>laskutaide</span></a></p>
Risto A. Paju<p>One more simple geodesic himmeli: a snub cube with the square faces augmented into pyramids. 30 vertices, 84 edges and Eulerian.</p><p>The "simple geodesic himmeli" is my own loose definition meaning</p><p>(a) Triangular faces only for a stable himmeli structure<br />(b) At most 2 different edge lengths for a simple construction<br />(c) Reasonably symmetric/balanced look<br />(d) Nearly equilateral triangles for symmetry/balance</p><p>The common way to make geodesic polyhedra starts with a Platonic solid (usually icosahedron, sometimes octahedron), splits the edges/faces evenly, and normalizes the vertices to a sphere. For my "simple" criteria, a useful alternative is to start with an Archimedean solid that contains triangles and one other kind of face, and split/augment those other faces into triangles. In many cases, the result is basically equivalent to a regular geodesic; in the present case, a {3,4+}_2,1. I think the edge lengths might be a bit different from the regular geodesic construction, but here the Archimedean starting point ensures that we only need 2 different lengths.</p><p>As I've noted earlier, Eulerian graphs enable himmeli constructions with a single thread and no backtracking. For my first nontrivial himmelis I used software to plan the route, but I now mostly just go ahead with intuition. You just need a little forward thinking when you're nearing the finish line, as it's easier to hit a dead end there. So by making himmelis you can learn an intuitive algorithm for Eulerian cycles, and a better understanding of polyhedra in general.</p><p><a href="https://mathstodon.xyz/tags/himmeli" class="mention hashtag" rel="tag">#<span>himmeli</span></a> <a href="https://mathstodon.xyz/tags/puzuri" class="mention hashtag" rel="tag">#<span>puzuri</span></a> <a href="https://mathstodon.xyz/tags/strawart" class="mention hashtag" rel="tag">#<span>strawart</span></a> <a href="https://mathstodon.xyz/tags/snubcube" class="mention hashtag" rel="tag">#<span>snubcube</span></a> <a href="https://mathstodon.xyz/tags/geodesichimmeli" class="mention hashtag" rel="tag">#<span>geodesichimmeli</span></a> <a href="https://mathstodon.xyz/tags/geodesicsphere" class="mention hashtag" rel="tag">#<span>geodesicsphere</span></a> <a href="https://mathstodon.xyz/tags/geodesicpolyhedron" class="mention hashtag" rel="tag">#<span>geodesicpolyhedron</span></a> <a href="https://mathstodon.xyz/tags/polyhedron" class="mention hashtag" rel="tag">#<span>polyhedron</span></a> <a href="https://mathstodon.xyz/tags/geometricart" class="mention hashtag" rel="tag">#<span>geometricart</span></a> <a href="https://mathstodon.xyz/tags/algorithmicart" class="mention hashtag" rel="tag">#<span>algorithmicart</span></a> <a href="https://mathstodon.xyz/tags/algorist" class="mention hashtag" rel="tag">#<span>algorist</span></a> <a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#<span>mathart</span></a> <a href="https://mathstodon.xyz/tags/laskutaide" class="mention hashtag" rel="tag">#<span>laskutaide</span></a></p>
Risto A. Paju<p>Continuing on my geodesic himmeli series, here are two specimens using octahedral/cubic symmetry, which I haven't used much. The vertex/edge/face counts are 14/36/24 and 18/48/32.</p><p>The first one is a tetrakis hexahedron, but not the Catalan solid, as the vertex positions are normalized to a sphere. You can also regard it as a cube with the faces augmented into pyramids. I think its Wenninger notation is {3,4+}_1,1.</p><p>The second one is a simple 2-frequency division of an octahedron, normalized to a sphere, or {3,4+}_2,0 in Wenninger's notation. Alternatively, it's a cuboctahedron with the square faces augmented into pyramids.</p><p>(Video posting doesn't seem to work, so here's a link to one: <a href="https://youtu.be/rELAPCx6ako" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">youtu.be/rELAPCx6ako</span><span class="invisible"></span></a>)</p><p><a href="https://mathstodon.xyz/tags/himmeli" class="mention hashtag" rel="tag">#<span>himmeli</span></a> <a href="https://mathstodon.xyz/tags/puzuri" class="mention hashtag" rel="tag">#<span>puzuri</span></a> <a href="https://mathstodon.xyz/tags/strawart" class="mention hashtag" rel="tag">#<span>strawart</span></a> <a href="https://mathstodon.xyz/tags/geodesichimmeli" class="mention hashtag" rel="tag">#<span>geodesichimmeli</span></a> <a href="https://mathstodon.xyz/tags/geodesicsphere" class="mention hashtag" rel="tag">#<span>geodesicsphere</span></a> <a href="https://mathstodon.xyz/tags/geodesicpolyhedron" class="mention hashtag" rel="tag">#<span>geodesicpolyhedron</span></a> <a href="https://mathstodon.xyz/tags/tetrakishexahedron" class="mention hashtag" rel="tag">#<span>tetrakishexahedron</span></a> <a href="https://mathstodon.xyz/tags/polyhedron" class="mention hashtag" rel="tag">#<span>polyhedron</span></a> <a href="https://mathstodon.xyz/tags/geometricart" class="mention hashtag" rel="tag">#<span>geometricart</span></a> <a href="https://mathstodon.xyz/tags/algorithmicart" class="mention hashtag" rel="tag">#<span>algorithmicart</span></a> <a href="https://mathstodon.xyz/tags/algorist" class="mention hashtag" rel="tag">#<span>algorist</span></a> <a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#<span>mathart</span></a> <a href="https://mathstodon.xyz/tags/laskutaide" class="mention hashtag" rel="tag">#<span>laskutaide</span></a></p>
Risto A. Paju<p>As a by-product of a failed idea, here's another simple geodesic himmeli: a truncated tetrahedron whose hexagonal faces are augmented into pyramids. With 16 vertices, 42 edges and 28 faces, it goes between the icosahedron and the pentakis dodecahedron in my geodesic series.</p><p>The tetrahedral symmetry is neither ideal nor typical for geodesic polyhedra — for a nice sphere, you'd usually start with an icosahedron. Incidentally, it does have 12 pentavalent vertices like the icosahedral ones, but they are not evenly distributed, so I'm not sure if there's a Wenninger notation for this. In any case, if you want a geodesic polyhedron with 16 vertices, here's a way to do it.</p><p><a href="https://mathstodon.xyz/tags/himmeli" class="mention hashtag" rel="tag">#<span>himmeli</span></a> <a href="https://mathstodon.xyz/tags/puzuri" class="mention hashtag" rel="tag">#<span>puzuri</span></a> <a href="https://mathstodon.xyz/tags/strawart" class="mention hashtag" rel="tag">#<span>strawart</span></a> <a href="https://mathstodon.xyz/tags/geodesichimmeli" class="mention hashtag" rel="tag">#<span>geodesichimmeli</span></a> <a href="https://mathstodon.xyz/tags/geodesicsphere" class="mention hashtag" rel="tag">#<span>geodesicsphere</span></a> <a href="https://mathstodon.xyz/tags/geodesicpolyhedron" class="mention hashtag" rel="tag">#<span>geodesicpolyhedron</span></a> <a href="https://mathstodon.xyz/tags/polyhedron" class="mention hashtag" rel="tag">#<span>polyhedron</span></a> <a href="https://mathstodon.xyz/tags/geometricart" class="mention hashtag" rel="tag">#<span>geometricart</span></a> <a href="https://mathstodon.xyz/tags/algorithmicart" class="mention hashtag" rel="tag">#<span>algorithmicart</span></a> <a href="https://mathstodon.xyz/tags/algorist" class="mention hashtag" rel="tag">#<span>algorist</span></a> <a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#<span>mathart</span></a> <a href="https://mathstodon.xyz/tags/laskutaide" class="mention hashtag" rel="tag">#<span>laskutaide</span></a></p>
Risto A. Paju<p>Another step in my series of geodesic Himmelis: a snub dodecahedron with the pentagons augmented into pyramids. Like my previous geodesics, this too can be made with just 2 different edge lengths. It is also known as {3,5+}_2,1 and has a total of 210 edges.</p><p><a href="https://mathstodon.xyz/tags/himmeli" class="mention hashtag" rel="tag">#<span>himmeli</span></a> <a href="https://mathstodon.xyz/tags/puzuri" class="mention hashtag" rel="tag">#<span>puzuri</span></a> <a href="https://mathstodon.xyz/tags/strawart" class="mention hashtag" rel="tag">#<span>strawart</span></a> <a href="https://mathstodon.xyz/tags/geodesicsphere" class="mention hashtag" rel="tag">#<span>geodesicsphere</span></a> <a href="https://mathstodon.xyz/tags/geodesicpolyhedron" class="mention hashtag" rel="tag">#<span>geodesicpolyhedron</span></a> <a href="https://mathstodon.xyz/tags/polyhedron" class="mention hashtag" rel="tag">#<span>polyhedron</span></a> <a href="https://mathstodon.xyz/tags/geometricart" class="mention hashtag" rel="tag">#<span>geometricart</span></a> <a href="https://mathstodon.xyz/tags/algorithmicart" class="mention hashtag" rel="tag">#<span>algorithmicart</span></a> <a href="https://mathstodon.xyz/tags/algorist" class="mention hashtag" rel="tag">#<span>algorist</span></a> <a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#<span>mathart</span></a> <a href="https://mathstodon.xyz/tags/laskutaide" class="mention hashtag" rel="tag">#<span>laskutaide</span></a></p>
Åsa Maria Hedberg - artist<p>Tankehjälp:<br>Behöver några hundra tunna korta rör. Tänk tjugocentimetersbitar, kan kapa själv förstås. Ska klara utomhuset. Bambu hade varit toppen men är för dyrt. PVC hade funkat men går bort pga plast och framförallt för dyrt. Halm är aningen för klent och smalt. </p><p>Finns det någon gratisvariant jag inte tänkt på?<br><a href="https://mastodon.art/tags/sl%C3%B6jd" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>slöjd</span></a> <a href="https://mastodon.art/tags/himmeli" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>himmeli</span></a> <a href="https://mastodon.art/tags/flaggst%C3%A5ng" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>flaggstång</span></a></p>
Åsa Maria Hedberg - artist<p>Fick en så enormt bra idé inför uppstart på textilslöjden på tisdag - onsdag!!</p><p>Skulle bara behöva få tag i en massa slöjdhalm eller ännu hellre ihåliga bambupinnar...<br><a href="https://mastodon.art/tags/flaggst%C3%A5ng" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>flaggstång</span></a> <a href="https://mastodon.art/tags/rockring" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>rockring</span></a> <a href="https://mastodon.art/tags/himmeli" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>himmeli</span></a></p>
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