论文标题
改进的折纸挤压中的平板3D小工具完全与常规金字塔支持的3D小工具完全兼容
Improved flat-back 3D gadgets in origami extrusions completely downward compatible with the conventional pyramid-supported 3D gadgets
论文作者
论文摘要
折纸挤出是使用3D小工具在平坦纸的中间折叠的3D对象，这些小工具会创建具有固体角度的面孔。在本文中，我们专注于3D小工具，该小工具会形成与周围纸的顶部面，两个侧面共享山脊，并带有两个外向的简单褶皱，其中一个简单的褶皱是一对山褶皱和一个山谷褶皱。有两种类型的3D小工具。一种是带有三角形金字塔的传统类型的3D小工具，从内部支撑两侧。另一种是我们以前的论文中提出的较新的3D小工具，它在几个方面改善了传统的小工具：它们在环境纸的上方有平坦的背面，侧面面之间没有缝隙；它们不太干扰相邻的小工具，因此我们可以一次使挤压更高。它们与常规的构造兼容；他们有一个修改后的扁平小工具，用于重复，不会干扰相邻的小工具。在某些条件下，它们即将发出的褶皱的角度可以改变。但是，在某些情况下，我们可以在无法以前的小工具时应用传统的小工具。本文的目的是改善我们以前的3D小工具，使其完全与传统的小工具完全兼容，从某种意义上说，任何传统的小工具都可以被我们改进的小工具取代，而我们的改进的小工具则具有相同的外向褶皱，但相反并不总是可能。更确切地说，我们证明，对于任何给定的常规3D小工具，都有与之兼容的无限改进的3D小工具，并且可以将常规的3D小工具替换为这些3D小工具，而不会影响任何其他常规3D小工具。另外，我们看到我们的改进的3D小工具使上述所有优势都比传统的优势。
An origami extrusion is a folding of a 3D object in the middle of a flat piece of paper, using 3D gadgets which create faces with solid angles. In this paper we focus on 3D gadgets which create a top face parallel to the ambient paper and two side faces sharing a ridge, with two outgoing simple pleats, where a simple pleat is a pair of a mountain fold and a valley fold. There are two such types of 3D gadgets. One is the conventional type of 3D gadgets with a triangular pyramid supporting the two side faces from inside. The other is the newer type of 3D gadgets presented in our previous paper, which improve the conventional ones in several respects: They have flat back sides above the ambient paper and no gap between the side faces; they are less interfering with adjacent gadgets so that we can make the extrusion higher at one time; they are downward compatible with conventional ones if constructible; they have a modified flat-back gadget used for repetition which does not interfere with adjacent gadgets; the angles of their outgoing pleats can be changed under certain conditions. However, there are cases where we can apply the conventional gadgets while we cannot our previous ones. The purpose of this paper is to improve our previous 3D gadgets to be completely downward compatible with the conventional ones, in the sense that any conventional gadget can be replaced by our improved one with the same outgoing pleats, but the converse is not always possible. To be more precise, we prove that for any given conventional 3D gadget there are an infinite number of improved 3D gadgets which are compatible with it, and the conventional 3D gadget can be replaced with any of these 3D gadgets without affecting any other conventional 3D gadget. Also, we see that our improved 3D gadget keep all of the above advantages over the conventional ones.
