当我渐渐睡着时，我还在想象在一个四维球体周围怎样构造一个超正方体。
I was trying to visualize how to construct a tesseract around a four-dimensional sphere when I drifted away to sleep.
这个方程能消除物体间的差异，并能把物件精确的变形为一个统一的三维球体。
The equation smoothes out the irregularities of an object, transforming it mathematically into something that looks like a uniform three-dimensional sphere.
他在1904年就提出猜想：在四维空间，所有单连通的封闭三维面都能转化为一个三维球体。
His conjecture, made in 1904, was that in this four-dimensional world, all closed three-dimensional surfaces that are simply connected could be transformed to look like a three-dimensional sphere.
1956年，米尔·诺尔证明了奇异的7维球体的存在，其他数学家立刻认识到这一成果是时代的杰作。
In 1956, Milnor produced a result that other mathematicians immediately recognized as a masterpiece for the ages: proving the existence of "exotic" 7-dimensional spheres.
然而米尔·诺尔构造了一个7维球体(实际上是28个维度)，这个球体非常扭曲，无需通过创建拐角和折叠来解析。
But Milnor constructed a seven-dimensional sphere (in fact, exactly 28 of them) that is too badly twisted to be unscrambled without creating corners and folds.
用数学语言表达就是“每个单连通3重（3-manifold）封闭体与一个三维球体同胚（homeomorphic）。”
As mathematicians say, “Every simply connected closed 3-manifold is homeomorphic to a 3-sphere.”
用数学语言表达就是“每个单连通3重（3-manifold）封闭体与一个三维球体同胚（homeomorphic）。”
As mathematicians say, "Every simply connected closed 3-manifold is homeomorphic to a 3-sphere."
这样的2维宇宙模型就像是球体，不可能出现平行的测地线(测地线是曲面上的直线)——两直线必然在某点相交。
A two-dimensional model of such a universe would look like a sphere. It's impossible to have parallel geodesics (straight lines on a curved surface) — the two lines will cross at some point.
来自Murphy的一个旋转的三维棋盘球体。
在他身后播放的视频中可以看到一个球体的大型三维图像。
Large three-dimensional images of a globe can be seen on a video that plays behind him.
因此，对于高硬度，布氏硬度计和维氏硬度计或洛氏硬度计的检测值之间的关系就会受到所采用的球体的影响。
At high hardness, therefore, the relationship between Brinell and Vickers or Rockwell scales is affected by the type of ball used.
本文分别给出了使在无穷维欧氏空间中球体和球面具有有限的，但又不是无穷小的测度的半径集合。
In this paper, we give the sets of radii which make the measures of balls and spheres finite but not infinitesimal respectively, in a Euclidean space of infinite dimension.
使用高斯-博内定理， 史蒂芬·霍金证明一个（四维）黑洞活动视界的拓扑结构是2个球体。
Using the Gauss-Bonnet theorem, Stephen Hawking proved that the topology of the event horizon of a (four dimensional) black hole is a 2-sphere.
本文应用零方程模型和二方程模型计算了椭球体有攻角绕流对称面上的三维湍流边界层。
Revised method is proposed for calculation of three dimensional turbulent boundary layers on the surface of symmetry plane of a flow around ellipsoid at incidence.
为了实现对球体三维位置的动态实时检测，提出一种基于预测的椭圆曲线快速搜索算法。
In order to realize dynamic real time detection of spherical 3d location, a fast searching algorithm for elliptical curve based on prediction is proposed.
球体是三维体。
最后构建全维状态观测器，实现对球体目标的跟踪。
Finally, based on full state observer theory, the ball can be tracked.
碟片里以三维立体空间介绍许多形状，比如立方体(对正方形)和球体(对圆形)。
The video shows the 3-dimensional versions of many shapes, such as the cube (for the square), and the sphere (for the circle).
建立了生成智能虚拟控件的环境与平台的系统模型、椭球体三维数据场可视化模型。
It builds up the model of the environment and platform where the intelligent virtual controls are formed and the ellipsoid tridimentional data field visualization model.
在模型中运用移动的三维双椭球体热源作为输入热源来模拟焊接过程，最终获得了焊接温度场和残余应力场。
The temperature field and residual stress field were obtained by using 3-D double-ellipsoidal heat source model to simulate welded process.
在模型中运用移动的三维双椭球体热源作为输入热源来模拟焊接过程，最终获得了焊接温度场和残余应力场。
The temperature field and residual stress field were obtained by using 3-D double-ellipsoidal heat source model to simulate welded process.
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