几个关键点：
1、泛正方体
二维中为方形：
-a<x<a , -a<y<a
三维中为正方体
-a<x<a , -a<y<a, -a<z<a
四维中为超立方体
-a<x<a,-a<y<a,-a<z<a,-a<m<a
可理解为我们的世界为m为0点处的四维世界，由于上述超立方体中x,y,z取值范围与m无关，当该超体与我们世界相交，即m取值在-a到a之中时，其他三维取值范围不变，在我们世界中始终未一个立方体。现通过正交变换对立方体进行旋转变换：
x=1,0,0,0
y=0,1,0,0
z=0,0,1,0
m=0,0,0,1
v1 = a1x+b1y+c1z+d1m
v2 = a2x+b2y+c2z+d2m
v3 = a3x+b3y+c3z+d3m
v4 = a4x+b4y+c4z+d4m
v1,v2,v3,v4两两正交
可得其中一解为
1,1,1,1
1,1，-1，-1
1，-1，1,-1
1，-1，-1,1
得表达式
-a<x+y+z+m<a ,-a< x+y-z-m<a, -a< x-y+z-m<a, -a< x-y-z+m<a
m取值在-a到a之中时，其他三维取值发生改变
以下为源代码（未经整理）：
演示地址
var container;
var renderer;
var scene;
var camera;
var group;
var i = 0;
var t = -10;
var swi_tch = 1;
var daz = 0.2;
function circle(){
scene.remove(group);
requestAnimationFrame(circle);
camera.lookAt(scene.position);
persSuperCube(t);
if(swi_tch==1){
t = t + 0.5;
if(t>30){
swi_tch=0;
}
}
else{
t = t - 0.5;
if(t<-30){
swi_tch=1;
}
}
}
function onKeyDown(e){
//alert(e.keyCode);
if(e.keyCode==32){
circle();
}
}
function persSuperCube(t){
group = new THREE.Group();
var meshsss = [];
for(var i=0;i<24;i++)
for(var j=0;j<24;j++)
for(var k=0;k<24;k++)
{
if(meshsss[i]==null){
meshsss[i]=[];
}
if(meshsss[i][j]==null){
meshsss[i][j]=[];
}
var x=i-12;
var y=j-12;
var z=k-12;
var m=t-12;
if(x+y+z+m>-12 && x+y+z+m<12 && x+y-z-m>-12 && x+y-z-m<12 && x-y+z-m>-12 && x-y+z-m<12 && x-y-z+m>-12 && x-y-z+m<12){
meshsss[i][j][k]=new THREE.Mesh(
new THREE.BoxGeometry(2, 2, 2 ),
new THREE.MeshNormalMaterial( { overdraw: 0.5 } )
);
meshsss[i][j][k].position.x = 2*x;
meshsss[i][j][k].position.y = 2*y;
meshsss[i][j][k].position.z = 2*z;
meshsss[i][j][k].matrixAutoUpdate = false;
meshsss[i][j][k].updateMatrix();
group.add(meshsss[i][j][k]);
}
}
scene.add(group);
renderer.render(scene,camera);
}
function init(){
//document.addEventListener( 'keydown', onKeyDown, false );
container = document.createElement( 'div' );
document.body.appendChild( container );
group = new THREE.Group();
camera = new THREE.PerspectiveCamera( 45, window.innerWidth / window.innerHeight, 1, 10000 );
camera.position.x = 140*Math.sin(Math.PI/4);
camera.position.y = 140*Math.cos(Math.PI/4);
camera.position.z = 150;
camera.rotation.x = -Math.atan(camera.position.x/camera.position.z);
camera.rotation.y = Math.atan(camera.position.y/camera.position.z);
camera.rotation.x = -0.25 * Math.PI;
camera.rotation.y = 0.25 * Math.PI;
scene = new THREE.Scene();
scene.add(camera);
var x, y,z=0;
var meshsss = [];
for(var i=0;i<40;i++)
for(var j=0;j<40;j++)
for(var k=0;k<40;k++){
if(meshsss[i]==null){
meshsss[i]=[];
}
if(meshsss[i][j]==null){
meshsss[i][j]=[];
}
x=i-20;
y=j-20;
z=k-20;
if(x+y+z>-15 && x+y+z<15 && x-y+z>-15 && x-y+z<15 && -x+y+z>-15 && -x+y+z<15 && -x-y+z>-15 && -x-y+z<15){
meshsss[i][j][k]=new THREE.Mesh(
new THREE.BoxGeometry(2, 2, 2 ),
new THREE.MeshNormalMaterial( { overdraw: 0.5 } )
);
meshsss[i][j][k].position.x = 2*x;
meshsss[i][j][k].position.y = 2*y;
meshsss[i][j][k].position.z = 2*z;
meshsss[i][j][k].matrixAutoUpdate = false;
meshsss[i][j][k].updateMatrix();
group.add(meshsss[i][j][k]);
}
}
scene.add(group);
var meshx = new THREE.Mesh(
new THREE.BoxGeometry(200, 1, 1 ),
new THREE.MeshNormalMaterial( { overdraw: 0.5 } )
);
meshx.position.x = 100;
//meshx.rotation.z = 0.25 * Math.PI;
meshx.matrixAutoUpdate = false;
meshx.updateMatrix();
scene.add(meshx);
var meshy = new THREE.Mesh(
new THREE.BoxGeometry(1, 200, 1 ),
new THREE.MeshNormalMaterial( { overdraw: 0.5 } )
);
meshy.position.y = 100;
//meshy.rotation.z = 0.25 * Math.PI;
meshy.matrixAutoUpdate = false;
meshy.updateMatrix();
scene.add(meshy);
var meshz = new THREE.Mesh(
new THREE.BoxGeometry(1, 1, 200 ),
new THREE.MeshNormalMaterial( { overdraw: 0.5 } )
);
meshz.position.z = 100;
//meshz.rotation.x = 0.25 * Math.PI;
meshz.matrixAutoUpdate = false;
meshz.updateMatrix();
scene.add(meshz);
renderer = new THREE.CanvasRenderer();
renderer.setClearColor( 0xffffff );
//renderer.setPixelRatio( window.devicePixelRatio );
renderer.setSize( window.innerWidth, window.innerHeight );
container.appendChild( renderer.domElement );
renderer.render(scene,camera);
circle();
}
