Dimension · How Many Numbers Describe It

01 · 描述一件商品01 · Describe one product

知道颜色后，再加大小和价格，是数值变大了，还是描述位置变多了？ After color, adding size and price—did a value grow, or did the description gain positions?

023D 维度升阶台3D Dimension Builder

逐轴增加型Add one axis

每次只增加一个独立特征Add one independent feature at a time

从一个点开始。点击“增加一维”，看场景沿新轴生长，依次成为线、平面和立方体；相机也会拉远并转入中景。 Start from a point. Add one dimension and watch the scene grow into a line, plane, and cube while the camera pulls into a useful mid-shot.

轴 1 · 颜色Axis 1 · color0.72

轴 2 · 大小Axis 2 · size0.46

轴 3 · 价格Axis 3 · price0.81

固定种子Fixed seed20260722

教学模拟值，非真实模型输出simulated for teaching — not real model output

当前维度Dimensions

独立位置independent positions

表示形状Representation

—

几何直觉geometric intuition

最大数值Largest value

—

不等于维度not dimension

商品 AProduct A

1 单位参照1-unit reference

拖动旋转 · 滚轮缩放 · 松手后恢复缓转Drag to orbit · wheel to zoom · idle rotation resumes

正在加载 3D；若此提示持续显示，需要 WebGL 或 three.js 未加载。 Loading 3D. If this remains visible, WebGL is required or three.js did not load.

03维度不是玄学：只数独立位置Dimension Is a Count of Independent Positions

0D → 1D → 2D → 3D

一个点A point

还没有可变化的数值位置，只能停在原点。With no varying numeric position, the object stays at the origin.

一条线A line

只用颜色位置，一个数就能在线上定位商品。Color alone uses one number to locate the product on a line.

一个平面A plane

颜色与大小彼此独立，需要横纵两个坐标。Color and size are independent, requiring x and y.

一个立方体A cube

价格再增加一个方向；现在可从真实相机绕着体积观察。Price adds a third direction, so a real camera can orbit the volume.

04最小机制：每加一个独立属性，维度加一Minimal Mechanism: One Independent Attribute Adds One Dimension

d = |features|

x ∈ R^d, d = |F|

F 是已选独立特征集合。上方每点击一次加号，|F| 与 d 同时增加 1。 F is the selected independent feature set. Each click above increases both |F| and d by one.

增加一个轴Add one axis 多问一个独立问题Ask one independent question

增加一个位置Add one position 对象多一个坐标The object gains a coordinate

维度 +1Dimension +1 与数值大小无关Independent of magnitude

05边界实验：把价格放大一万倍Boundary Test: Multiply Price by Ten Thousand

主动制造失败Create a failure

数值暴涨，维度会涨吗？If a value explodes, does dimension grow?

点击后自动进入 3D，并把价格从 0.81 改成 8100。最大数值会变，轴数不会变。 The test builds 3D and changes price from 0.81 to 8100. The largest value changes; the axis count does not.

尚未执行。高维说的是几个数，不是数有多大。Not run yet. High-dimensional counts values; it does not measure their size.

✗ 价格 10000 比价格 10 更高维。A price of 10000 is higher-dimensional than 10. → ✓ 两者都只占价格这一个位置。Both occupy the same single price position.

✗ 三个相关副本就是三个独立维度。Three duplicate columns give three independent dimensions. → ✓ 维度强调彼此独立的数值位置。Dimension concerns independent numeric positions.

✗ 画不出四维，就无法计算四维。If 4D cannot be drawn, it cannot be computed. → ✓ 数组仍可保存并计算任意有限维度。Arrays can store and compute any finite dimension.

06 · 一句话带走06 · One line to keep

维度是描述对象所需的独立数值位置个数；点击增加轴，表示从点到线、面再到体，计数随位置增加而不是随数值变大。 Dimension is the number of independent numeric positions required. Add axes to move from point to line, plane, and volume; the count follows positions, not value size.