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← 判断力与美学 ← Judgment & Aesthetics
SAE 判断力与美学 · 余项之美
SAE Judgment & Aesthetics · Beauty of the Remainder
2026-09-18

商店里没有双曲线规

They Don't Sell Hyperboloid Compasses

Han Qin (秦汉)

Jordan Matelsky 做计算神经科学,博客叫 Code, Coffee, & Cortex。七月十八日那篇的头一句是:我有一台笔式绘图仪。很好。就像有一台打印机,但更慢,而且更容易坏。机器是 eBay 上买的,到手就半坏,他随即把买价与新机价格之间的那笔差额,原封不动地花在了修它上面。修好之后他给自己立的规矩是:只用它做打印机做不出来、手也做不出来的东西。上一件成果是一批金墨明信片,印的是一具禽龙化石。这一次他决定用它刻全息图——不是激光全息,是刻痕全息:在一块透明塑料上刻出一片曲率经过计算的细划痕,每道划痕把光反射成一条亮线,曲率决定你转头时那条亮线滑动的快慢,一堆划痕合起来就给出深度。成品在普通光线下什么都不是。要看见它,你得关掉所有的灯,举一支手电,把光直接反射进自己的眼睛。

他推导这套光学的方式值得单独记一笔。现成的权威解释是有的:William J. Beaty 二〇〇三年在 SPIE 的《Practical Holography XVII》上发过《Drawing holograms by hand》,Matelsky 自己在脚注里老实交代,他是看了 Grant Sanderson 讲全息的视频之后才找到那篇的。但他没有从那篇讲起。他选了一瓶特级初榨橄榄油,抹了满手,然后去摸自己的手机屏幕——「故意把手指抹上油再去碰手机屏幕,是一种相当离奇的体验。你试试!规矩都是编出来的。」油沿着指纹的脊线留下一片痕,光从某一段脊线上弹进眼睛,屏幕上于是浮起一道亮条;换一个角度,亮条沿着痕迹滑走。原理到此为止,没有别的了。这和 CD 表面那道彩虹披萨角是同一件事,和雨刷在挡风玻璃上刮出的那圈拖影也是同一件事。一个受过训练的人,手边就摆着一篇同行评议论文,却先去把橄榄油抹在手机上——这不是通俗化。这是他不肯直接接手别人已经凿好的那个形状。

真正的余项藏在一条脚注里,而且已经在那里躺了一百年。每一个要刻的点,正确的曲线是一段双曲面截线;Matelsky 把式子写了出来:x = d·tan(θ),y = d·sec(α)·(sec(θ)−1)。而在他之前,Beaty 和网上所有画手绘全息的人,用的都是圆弧近似。脚注四把理由写全了,一共两条:一,人通常只从很小的角度看;二,「商店里买得到画圆的圆规,但他们不做双曲线规。」第二条才是要害。那条曲线之所以一百年来是圆的,与光学无关,与被画的对象也无关——那是工具的形状爬到了对象身上,然后留在了那里。这是凿构循环最干净的一个切面:一次凿削所受的限制被记进了构,构一旦固化,限制就不再被当作限制,而被当作这门手艺本来的样子。Matelsky 拆掉它靠的不是新理论,是一台没有手腕的机器。他的原话是:「但我们用的是笔式绘图仪,所以形状的细节对我们来说无所谓。我们有一个机器人替我们画。」双曲线一直都在那里。圆才是一只手留下的余数,而这个余数被当了一百年的正文。

第二重余项在材料上,而且是双份的。一张刻痕全息图上没有任何被加上去的东西:没有颜料,没有乳剂,没有沉积物。你看见的每一道光,都是从「已经不在那里的材料」的断口上弹回来的。像完全由被刮掉的部分构成——这大概是「余项」这个词能落到最实处的一次。而承载它的材料本身也是一件遗留物。前两种材料都失败之后(下面再说),最后成的是旧 CD 塑料盒。他上 eBay 花几美元买了一批,本以为买的是新盒子,收到的是「一大箱又旧又脏的用过的」。他的反应是:「奇怪!不过……行吧,我接受。」脚注五还补了一句:这完全不怪卖家,而且再多花几美元,他本来可以买到「连碟一起的」。这句话请读慢一点。那只盒子存在的全部理由是保护里面那张碟;如今碟成了要另外加钱的选配,盒子成了主体,并且在一种与音乐、与数据、与一九八二年的任何设计意图都毫不相干的用途上,被证明是目前最好的材料——因为它够硬、够平、够便宜、够多。一个容器活得比它的内容物更久,于是第一次被看见。

两次失败也该记下来,因为它们是这件事目前唯一的实验记录。第一次用文具店的透明覆膜片:太软,刻的时候会跟着动,而且太波浪,反射面和视线不共面。第二次用小学时画画那种打蜡彩纸,拿木签去刻:还行,但他需要的曲线数量太大,纸撑不住,最后是撕裂加起皱。真正能刻的工具,是他父亲给的一套钩针挑针(他记得是 Harbor Freight 的),其中最尖的那根直挑——视频里橙色柄的那根——在塑料盒上极好用,而在纸上「彻底把它毁了」。材料的限度还有更要紧的一条:刻得越多,效果越好,但表面也越来越毛,采样密到一定程度,整片刻过的区域就变成「一大块哑光糊斑」。也就是说这件作品有一个最优点,越过那个点,继续劳动会把它抹掉。至于观看条件:光源必须是点光源,光源越宽,虚像的每个点就越宽,深度信息越少;最好的位置是点光源的主眩光刚好压在划痕顶端之上。他承认拍出来的东西「在镜头里有点让人失望」,但他一个人举着手电待在漆黑的衣柜里、像鸽子一样前后晃脑袋去看第一张成品的时候,那是 magical。相机在这里不是记录不足,是结构性地记录不了——这件作品的全部内容,就是观看者移动时发生的事。

然后是名字。文中有一节的标题就叫《这些真的算全息图吗?》。他的回答不是搬定义,而是挑了一条他自己最喜欢的性质:哪怕切下很小一块,那一块里仍然保有完整的场景。他写:「这不是一个正式定义——这只是我最喜欢的那个关于全息图的事实。」接着他把命名权直接交了出去:「如果你坚持认为这些不是全息图,那就替我找一个更好的名字,你可以管它叫那个。我就是觉得它们挺酷的。」一件还没有名字、而且作者本人拒绝抢占命名的东西,正是余项在被收编之前的标准形态。Beaty 那篇论文末尾还有一段附言,Matelsky 说它让他起鸡皮疙瘩——所需工具如此简单,古人本可以用木器在凝固的松烟树脂面上画出这些像,只要他们知道这个诀窍;好几个古文明都深研过几何。「我在想,会不会有某些器物此刻正躺在某处的库房里,抛光面上带着一些看起来像是砂纸划痕的东西……」这是我见过的关于余项最锋利的一句:一整类对象可能早已存在、早已入藏、早已被编目为「有磨损划痕」,只因为没有人拿一支点光源去照过它们。没有名字,所以没有对象。为什么现在看比以后看更要紧,看它此刻被放在哪里就知道了——这不是作品发布,是一篇个人博客,标签栏写着 pen-plotter、art、holography、optics、blender,上一篇是禽龙化石明信片,脚注一里他交代当初买绘图仪本是想让它画水彩,后来发现有位叫 Licia He 的艺术家把这件事做得很好,「然后我就不好意思了,一直没做」。他甚至不肯给自己买一支金刚石刻针——「我可不想在一个还不够酷的项目上花那种钱!你有闲着的想送我吗?」一个作者公开把自己的项目叫作 not-cool-yet,这是一枚时间戳。等这件事拿到名字——「绘图仪全息」、一个长青生成系列、一场「另类工艺」工作坊——手电和衣柜就会从叙述里消失,双曲线会变成一个参数,旧塑料盒会变成一种材质选项,剩下的是又一批可以拍照上传的完成品。此刻它还只是一个人把橄榄油抹在手机上,而且还没有人决定它是什么。

blog.jordan.matelsky.com ↗

Jordan Matelsky works in computational neuroscience and writes a blog called Code, Coffee, & Cortex. His July 18th post opens: I have a pen-plotter! It's great. It's like having a printer but slow and it breaks more often. He bought the machine on eBay, it arrived sort of broken, and he immediately spent the difference between what he paid and the price of a new one on fixing it. Having fixed it, he set himself a rule: use it only for things that can't be made with a printer or by hand. The last result was a run of gold-ink postcards of an Iguanodon fossil. This time he decided to etch holograms — not laser holograms, scratch holograms: a field of fine, precisely curved grooves cut into clear plastic, each groove throwing back a single line of glare, the curvature of the groove setting how fast that glare travels when you move your head, the whole field adding up to depth. Under ordinary light the finished object is nothing at all. To see it you turn off every light in the room, hold up a flashlight, and reflect it straight into your own eyes.

How he derives the optics deserves recording on its own. The authoritative account was sitting right there: William J. Beaty published "Drawing holograms by hand" in SPIE's Practical Holography XVII in 2003, and Matelsky admits in a footnote that he found it after watching Grant Sanderson's video on holography. He does not start from it. He starts by selecting a nice extra virgin olive oil, greasing his fingers, and touching his phone screen — "It was a truly bizarre experience deliberately smearing my fingers with oil and then touching my phone screen. Try it! Rules are made up!" The oil leaves ridges along the fingerprint; light finds a path off one stretch of ridge into your eye; a highlight surfaces on the glass; move, and the highlight slides along the smear. That is the entire principle, and there is nothing else to it. It is the same thing as the rainbow pizza-slice on a CD, the same thing as wiper streaking on a windshield. A trained person with a peer-reviewed paper at hand going first to the olive oil is not popularization. It is a refusal to take delivery of a shape somebody else has already chiselled.

The real remainder is in a footnote, and it has been lying there for a century. For each point you want to render, the correct curve is a hyperboloid section; Matelsky gives the equations: x = d·tan(θ), y = d·sec(α)·(sec(θ)−1). Everybody before him — Beaty and every hand-drawn hologram online — used a circular arc instead. Footnote 4 states the reasons in full, and there are two: one, you usually look from a small angle; two, "they make circular compasses that you can buy at a store but they don't make hyperboloid ones." The second reason is the one that matters. The curve has been a circle for a hundred years for reasons that have nothing to do with optics and nothing to do with the object being drawn. It is the shape of the instrument, climbed onto the object and left there. This is as clean a cross-section of the chisel-construct cycle as you will find: a constraint on one act of cutting gets recorded into the construct, and once the construct sets, the constraint stops reading as a constraint and starts reading as what the craft simply looks like. What Matelsky breaks it with is not a new theory. It is a machine with no wrist. In his words: "But we're using a pen plotter, so the specifics of the shape are immaterial to us. We have a robot to do it for us." The hyperbola was always there. The circle was the residue of a hand — and the residue has been read as the text for a hundred years.

The second remainder is in the material, twice over. Nothing on a scratch hologram has been added: no pigment, no emulsion, no deposit. Every line of light you see comes off the broken edge of material that is no longer there. The image consists entirely of what was scraped away — about as literal as the word remainder ever gets. And the substrate is itself a leftover. After two materials failed (below), what finally worked was an old CD jewel case. He bought a batch on eBay for a few dollars, assuming he was buying new ones, and received "a huge box of old and dirty used ones." His reaction: "Weird! But also… Okay I guess!" Footnote 5 adds that this was totally not the seller's fault, and that for just a few dollars more he could have bought the CDs in the cases. Read that slowly. The case exists for one reason, which is to protect the disc inside it; the disc is now the paid upgrade, the case is the object, and in a use that has nothing to do with music, with data, or with any design intent from 1982, it turns out to be the best available material — because it is hard enough, flat enough, cheap enough, and plentiful enough. A container has outlived its contents, and is seen for the first time.

The two failures are worth keeping, because they are the only experimental record this thing has. First, a clear plastic lamination sheet from an office supply store: too flexible, it moved while he etched, and too wavy to reflect light coplanar to the viewing angle. Second, the waxed coloured paper you drew on in elementary school, etched with a wooden stylus: alright, but the volume of curves he needed was more than the paper could take, and it tore and crumpled. The tool that actually cuts is a hooks-and-picks set his dad gave him, from Harbor Freight he thinks, of which the pokiest straight pick — the orange-handled one in the videos — works beautifully on jewel cases and "totally annihilated" the papers. There is a harder material limit still: the more etches he makes, the better the result, but also the rougher the surface, and at high enough sampling the etched region just becomes a big matte blob. Which is to say the work has an optimum, past which further labour erases it. As for viewing: the source must be a point source, because the wider the source the wider each virtual image point and the less depth survives; the illusion is best when the main glare sits just above the top of the etches. He admits the results are a bit underwhelming on camera — but sitting with a flashlight in his pitch black closet, moving his head around like a pigeon, the first working hologram was magical. The camera here is not under-documenting. It structurally cannot document, because the entire content of the work is what happens while the viewer moves.

Then there is the name. One section of the post is headed "Are these really holographic?" His answer is not a definition but a property he happens to like best: even a small cutout retains a complete copy of the scene. He writes: "This isn't a formal definition — it's just the fact about holograms that I like the most." And then he hands the naming rights away outright: "if you feel strongly that these are not holograms, then find for me a better name and you can call them that thing. I just think they're neat!" Something with no name yet, whose maker declines to claim the naming, is the standard form a remainder takes before it is absorbed. Beaty's paper ends with a postscript that Matelsky says gives him shivers — the required tools are so simple that ancient peoples could have drawn these images in hardened sooty resin pools with wooden tools, had they but known the trick, and several ancient civilizations were heavily into geometry. "I wonder if any artifacts exist in storage somewhere which appear to have some 'sandpaper marks' on their polished surfaces…" That is the sharpest sentence about remainders I have read: an entire class of objects may already exist, already be accessioned, already be catalogued as scuffed, purely because nobody has held a point source up to them. No name, therefore no object. As for why seeing this now matters more than later — look at where it currently sits. This is not a release; it is a personal blog post tagged pen-plotter, art, holography, optics, blender, following a post about dinosaur postcards, with a footnote admitting he got the plotter in the first place to make it paint watercolours, then found an artist named Licia He who does that really well, "and then I got self conscious and never did it." He won't even buy himself a diamond-tipped stylus: "I sure don't want to spend that kind of money on a not-cool-yet project! Do you have one laying around that you want to give me?" An author calling his own project not-cool-yet, in public, is a timestamp. The moment this gets a name — plotter holography, a long-form generative series, an "alt process" weekend workshop — the flashlight and the closet drop out of the telling, the hyperbola becomes a parameter, the used jewel case becomes a material option, and what is left is another batch of finished surfaces to photograph and upload. Right now it is still a man rubbing olive oil on his phone, and nobody has decided yet what it is.

blog.jordan.matelsky.com ↗