The 17-Klein Bottle - Numberphile

17 Klein Bottles become 1 - featuring Cliff Stoll and the glasswork of Lucas Clarke.
More links & stuff in full description below ↓↓↓
More Cliff videos: bit.ly/Cliff_Videos
More Klein Bottle videos: bit.ly/KleinBottles
Buy glassware from Cliff (if you dare): www.kleinbottle.com
CORRECTION (message via Cliff): At 43 seconds (recycling symbol) I goofed here. That's a single-half-twist. I'm properly one-sidedly chagrined.
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outr...
And support from Math For America - www.mathforamerica.org/
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Animation by Pete McPartlan
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[@numberphile]

More Cliff videos: bit.ly/Cliff_Videos More Klein Bottle videos: bit.ly/KleinBottles

[@RKrk-jj2li]

Wait a minute: how come it doesn't count, what the camera man said?

[@ipudisciple]

@@RKrk-jj2li The glass model is an attempt to portray an object that can't be embedded in 3 dimensions, so it has to go through itself in a few places. But it exists in abstract, and that's what the true Klein bottle is. The cameraman and you are talking about the embedding; Cliff is talking about the abstract object. You're both right, in different ways.

[@prefeitobear9209]

Math doesn't mean practical.

[@fuzzylumpkin8030]

Numberphile a real life mad scientist 🤪

[@edisoncarter3841]

The single Klien Bottle is a "half twist" so a 2nd would be a full twist (2 sided). Odd number of bottles would each be a half twist. Which makes sense since the single bottle is a mobius strip with edges stretched and sealed to form a single sheet. You could make any odd number of bottles without disrupting the single sided surface.

[@carlwitt7950]

Find someone who loves you as much as he loves Klein bottles.

[@TitaniumDruid]

BRB, dying alone

[@TheAlps36]

Or someone who can make them

[@karlboud88]

Like number 1000 had to do it

[@EmergentSea1]

Impossible.

[@boysimpledimple]

Eric Weinstein

[@MaxDiscere]

At this point he's just flexing arround with his Klein-Bottles

[@filthyjarod7808]

True

[@mirjanbouma]

Can you blame him? I would too if I had such awesome Klein bottles.

[@mathandcubes3741]

Mirjan Bouma me too

[@mathandcubes3741]

Especially the Möbius strip Klein bottle

[@maho7625]

ye weird flex bro

[@SgtElev3n]

the glass blower guy is the real MVP here

[@stuiesmb]

Elev3n we need an interview with this glass blower on Numberphile2

[@elijahshadbolt7334]

They mass produced 1000 regular klein bottles to make it affordable per bottle

[@superscatboy]

For real! The level of skill of the glassblower is literally awesome.

[@TintagelEmrys]

@@superscatboy and the patience

[@IndigoGollum]

I thought Cliff made them.

[@talkingscribe8898]

"I've already broken it once, so now I treat it with care," he says while spinning it like a pizza.

[@yazajag]

😂😃🍕

[@chubishroom9500]

If 'with care' is like that, imagine without care

[@RanoutofideazXD]

@@chubishroom9500 I was gonna say-

[@spaceguyreal]

@@RanoutofideazXD chaos

[@Marconius6]

"Cliff, it's 2020, should we use computer modeling to examine this idea of 17 linked Klein bottles?" "I'll pretend I didn't hear that. To the glassblowers!"

[@TrackpadProductions]

You should read what he said about the internet in his 1995 book, "Silicon Snake Oil". Did _not_ age well.

[@user-me7hx8zf9y]

@@TrackpadProductions He said it was his biggest blunder...

[@TrackpadProductions]

@Joseph Malone Basically that the internet was a fad that would never catch on. That online shopping would never happen, that people would never get news on computers, that books would never be read on mobile devices, stuff like that.

[@jengleheimerschmitt7941]

I hope the glassblower burned only charcoal and dung.

[@OrangeC7]

@@TrackpadProductions To be fair, at that time it totally could have seemed impossible.

[@SuryanIsaac]

This dude's enthusiasm is what the world needs

[@lx4302]

Suryan Isaac i need that enthusiasm

[@infatrump1]

If every nat teacher was like this id be way more interested in it

[@scottbaldwin3083]

He's also tenacious. He wrote a true-crime book about how he caught a hacker mucking around government computers in the early-mid 80's. Wasn't easy, but this guy figures stuff out!

[@rarelycomments]

@@scottbaldwin3083 lol what

[@nlinchmc9509]

I once had a neighbor that is like this about his things that are old. One of the main reasons we are still in contact after 10 years is his enthusiasm whenever we talk. We have been there (ca.150 km away on holiday) and planned to visit him for one hour. We ended up staying 4 hours so that it was already dark. We used about an hour to leave because it we started talking again and again. It was totally worth it!

[@Ojisan642]

“This is homeomorphic to a torus.” Holy cow, I actually understood that. Thanks, Numberphile!

[@8BitRip]

Slowly and surely I’m getting smarter watching these videos

[@succulent_pork3254]

Vsauce?

[@dmg4415]

I sort of start to understand. But his passion for these things are astounding.

[@sohenwei6937]

cow

[@Pedro13542]

@@succulent_pork3254 the squaring donuts video lol

[@jonclark2234]

Cliff’s students: There’s two sides to every argument, professor. Cliff: Well actually...

[@caniscerulean]

So a Klein bottle argument is when you actually agree with each other on an issue, but you are mistaking the same side opposite your plane as an opposing side?

[@22Hobbies]

@@caniscerulean *bickering* "Wait, so we're in agreement?" "...I suppose so." "...Lunch?" "Lunch, it is!"

[@StormTheSquid]

@@caniscerulean If I had a dollar for every time this has happened to me, I could buy a Klein bottle.

[@Niezow]

Just give the argument half a twist...

[@justaregulartoaster]

"Oh no not again! Evacuate!"

[@nani1039]

“YOU’RE CHEATING IF YOU GO BONK LIKE THAT!”

[@RKrk-jj2li]

I still don't understand why that's cheating

[@somerandomdragon558]

@@RKrk-jj2li A Klein bottle can only truly exist in 4 dimensions. In 3 dimensions, self intersections are not avoidable. It is an entirely different part of the bottle. If you would draw a line and go blonk like that and then unfold the Klein bottle (assuming you could somehow actually do that), the line would not be connected.

[@dave4148]

RK rk Because in the “real” object there’s no intersection so you would basically be jumping to a different part of the object

[@frostyusername5011]

@@RKrk-jj2li it's a 4 dimensional object, but in 3 dimensions it self intersects. That intersection is not *real*. When following the part that goes into the bottle, you can't jump back to the outside you must remain on the plane.

[@moth.monster]

Me when I get killed by the same scout for the fifth time in a single match of tf2

[@SeasOfCheese929]

I never would have guessed that two Klein bottles are homeomorphic to a torus.

[@MakusinMeringue]

and with the "Hole in a Hole in a Hole", that means we could get a similar object! It's all connected!!!!

[@KaliTakumi]

Same dude... That kind of hurts my brain lol

[@CharlieQuartz]

I'm not convinced that two connected Klein bottles are actually homeomorphic to a torus, since a strip with two half-twists is not homeomorphic to a normal loop. I would suspect that it would be analogous to a torus but with a couple of "twists" in its surface. In other words, the "twists" in two Klein bottles couldn't be worked out with normal topological operations to arrive at a 4R-embedded torus.

[@SocksWithSandals]

In a multiple choice of one answer, I would have guessed that wrong!

[@KaliTakumi]

@@CharlieQuartz A strip has edges. A taurus doesn't, how would it have twists?

[@mademedothis424]

5:50 You cannot convince me that this man's entire aesthetic isn't deliberately built on Back to the Future still shots.

[@mirjanbouma]

I'm pretty sure the character of that doctor wossname is based on this guy.

[@mademedothis424]

@@mirjanbouma Hah. I'm not sure the dates line up, but you know what? I'll gladly run with that.

[@qwertyTRiG]

@@mademedothis424 When we're dealing with time travel, who can say whether the dates line up?

[@MichaelSteeves]

When "Back to the Future" was released, Cliff was chasing down some of the first internet hackers. Read "The Cuckoo's Egg" to see his passion play out in words!

[@mirjanbouma]

@@mademedothis424 details! Insignificant details. 😁

[@ChrisSeltzer]

Shoutout to whoever does the graphics for these videos. The animations are extremely helpful in understanding the content and I personally really appreciate the effort.

[@JohnDixon]

Can we take a moment to appreciate the glass blower that made these? They're mind-blowing

[@Ricocossa1]

Pun intended?

[@Hydroxyium]

I meant 8:45

[@dreamsolutions3037]

Glass* blowing

[@otakuribo]

9:15 *has hands full "... and this hole here...." **bops it with his nose*

[@sohamsengupta6470]

That's the Cliff stol signature

[@marshmelows]

Protecc him!

[@itisALWAYSR.A.]

boop, except the b and p are connected in higher dimensions

[@boriswilsoncreations]

@@itisALWAYSR.A. so is *boop* a Klein bottle then? 😂

[@greenwaldian]

@@boriswilsoncreations No it's probably homeomorphic to a donut or somethin'

[@SailorSnap]

I am always so excited to see Cliff be excited about topology. And Lucas Clarke's glassblowing! So many creative and wonderful people together.

[@TheWolfboy180]

I love how angry Cliff gets when Brady asks why you can’t just use the self-intersections to get from one side to the other. Our 3D geometry is as alien to him as 4D geometry is to most of us

[@ipudisciple]

I don't think he's angry, but if he is, he's angry at the puny 3-dimensional world that he's forced to live in, not being able to make and hold the higher dimensional things that he can imagine.

[@cnadiajasmine]

Yeahh he looked so upset, like how could anyone ever even think of doing such a cheat??! Lol he's great

[@daddymuggle]

It's not just a cheat, it's impossible. Those self intersections don't exist on a real klein bottle.

[@YelDohan]

We need seven red lines all perpendicular, some with green ink and some transparent.

[@mebamme]

All the videos about Klein Bottles I've seen say they're awesome. They're all pretty one-sided.

[@RKrk-jj2li]

I like it.

[@N.I.R.A.T.I.A.S.]

Well done.

[@leefisher6366]

I have to admit that I'm into your argument... and out of it, at the same time.

[@jpanza]

take my like and leave

[@superksnowman1621]

This should’ve blown up way more 😂

[@gamebreakers2535]

The amount of passion cliff has for topology is more than my passion for literally anything on this planet

[@xpqr12345]

I get the impression that Cliff has the same amount of passion for a number of other topics. There are a couple of Numberphile videos in which he talks about eg a mechanical calculator he has (was it the Millionaire Machine?) and an early electronic calculator with an oscilloscope display and some sort of coiled spring memory. Both are well worth watching.

[@Xeridanus]

You should get that fixed.

[@bartsola8349]

@@xpqr12345 also the one with that city and the 7 bridges and I think there was one on a sliding rule or something like that? Great guy, very passionate

[@Triantalex]

false.

[@martinshoosterman]

When kliff calls someone nutty, you know that they are an extra special person.

[@Teck_1015]

Ahahaa. I love Cliff's enthusiasm. "Nahh!! You're cheating if you go like this!!" XD What a guy.

[@Cr42yguy]

8:45 the excitement to show us the mother of all Klein Bottles

[@Enceladus5280]

he's like an old hippie showing off a crazy bong collection

[@aiksi5605]

heHIheheheheheh...MEHEH

[@salty_slug]

like an evil scientist XD

[@roku_nine]

Kleinception bottle

[@randombrit13]

@@Enceladus5280 KLIEN BONG

[@bokkenka]

11:38 -- Are we sure Lucas isn't Cliff who traveled back in time through a cosmic-sized Klein bottle?

[@MegaAwesomeNick]

Cliff is actually Lucus's future self that has traveled back in time to share the secrets of Klein bottle technology that has caused a new scientific revolution. He just hasn't revealed his secret identity yet out of fear of creating a temporal anomaly.

[@n.l.4025]

But apparently the nuttiness of the Klein Bottle time traveling has caused him to lose his memory of his identity because he really doesn’t seem to know Lucas is his past self.

[@Cory_Springer]

Lucas and Cliff are the same, only separated by 1 Klein-Year.

[@marcelobraga3154]

"How many Klein bottles do you want Cliff?" Cliff: "YES"

[@LeventK]

For now, i only want 10 000, maybe more later...

[@Triantalex]

??..

[@Eleni_E]

Nobody gets as excited as a mathematician who’s explaining their favorite bit of maths. It’s so pure. I love it.

[@greggregson1365]

9:03 big time flexer has trouble finding normal Klein bottle

[@tuesdaymartesvr1942]

Biggy Cheese you are the ultimate flexer bro

[@Triantalex]

false.

[@lo1bo2]

4:50 - "Hot Diggity!" Haven't heard that one in a while.

[@rillloudmother]

The cousin that Doc Brown says is a bit eccentric...

[@yazajag]

You ain't lying. 🤔😃

[@MegaAwesomeNick]

I have never heard a voice of deeper happiness and contentment than this man unpacking a Mobius strip of Klein bottles.

[@hauntedmasc]

Cliff: *exists* Also Cliff: "Lucas is a real nut." Cliff Stoll is my hero :)

[@mathandcubes3741]

Me too

[@mathandcubes3741]

I love nerds like that since I can relate

[@Soken50]

they say it takes one to know one. Sounds like they were made to work together.

[@illustriouschin]

@@mathandcubes3741 Does this actually mean anything or are you just regurgitating a meme out of context?

[@JohnLeePettimoreIII]

Have you read "The Cuckoo's Egg"?

[@landonkryger]

0:40 The recycle sign only has 1 twist in it. If you look closely, one of them twists the opposite direction which cancels another twist, leaving it with just 1 twist.

[@SocksWithSandals]

Just looked that up. It's true. recycle symbol OK, Google wouldn't accept my Duck Duck Go link 😢

[@CharlieQuartz]

+

[@CharlieQuartz]

I cut a strip of paper out just to see this for myself. The fact that two of those twists cancel each other out and just redirect the strip is weird to see. Also a three-half-twist Mobius loop isn't nearly as pretty. I'm glad they went with the one-half-twist design.

[@jursamaj]

Yep. I was looking to see if anybody else spotted that.

[@AndrewPitmanator]

Shhh! Nobody tell Cliff.

[@EvilNightwolf]

8:54 That is one of the most awesome things i've seen for a while. It's a real piece of art. I wonder if you could do even more with a construct like that. Maybe colored glass, colored fluids or light.

[@MrMaxKeane]

Cliff: "Well I'll just head over to the glass shop." Glass shop owner: Sees cliff through window, "Hey, guys, the klein bottle guy is back."

[@word6344]

I like to imagine the glass shop owner made their own windows in this scenario

[@dhilboy231]

Numberphile: Uploads video with Cliff *Everyone liked that*

[@i_am_aladeen]

He is as entertaining as he is genius.

[@xSimonTan]

Reddit moment

[@i_am_aladeen]

@@xSimonTan I do :)

[@nausadge]

I have to watch his videos on mute with captions. Too much energy for me.

[@Marconius6]

0:00 Is this Cliff? 0:01 Of course it is!

[@carltonleboss]

Of course, Klein bottles are his forte!

[@dragatus]

The title of the video already gave it away.

[@austinbryan6759]

@@dragatus He was just making a joke about Cliffs obsession with Klein bottles

[@dragatus]

@@austinbryan6759 So was I. ;)

[@jobplace5842]

"nah, that's just a figment of our three dimensional universe." 3:00

[@cheeseisgreat24]

Cliff is my absolute favorite presenter on Numberphile, because he's got such a sincere enthusiasm that's absolutely infectious and voluminous.

[@talkingscribe8898]

2:32 "The A-side is different from the B-side." Anyone with a decent record collection could tell you that.

[@Ganliard]

There are music tapes that are a Möbius loop though

[@Triantalex]

false.

[@the_choco]

Always a pleasure to have Cliff on Numberphile, such an inspiring man

[@jobplace5842]

this man....... is my favorite. Klein bottles under his house, the millionaire machine.... someone should study this man as a cure for all things bad.

[@TheSentientCloud]

Cliff Stoll is literally the most adorable professor in existence. He gets so excited over math and I love that.

[@talkingscribe8898]

0:39 The top arrow on that Recyclable symbol is different, so the whole thing only had one half twist.

[@fss1704]

Talking Scribe yeah they draw it wrong apparently, i was going to say they didn't because the original is a mobius strip but then i rewinded and you were right.

[@jursamaj]

I'm glad I wasn't the only one to spot this…

[@user-rv9vk8by5i]

What on earth is the point?? This was clearly intentional - they could've just copypasted the same arrow. But nope, one twists the other way.

[@squeakybunny2776]

Finally... The comment i was looking for... Stared way too long at that thing thinking I was looking at it the wrong wat

[@SteveHodge]

That's the original design: two over arrows, one under arrow, so one half twist. Now many versions have all arrows the same resulting in 3 half-twists.

[@TheOfficialCzex]

I was considering purchasing the three-way Klein bottle, but, *clearly,* this is the optimal product...

[@Triantalex]

false.

[@shaggy0917]

"I treat it with delicacy and joy." I absolutely love Cliff!

[@oskare5289]

Cliff: let's add a third Me: three klein bottles isn't too bad Me: looks at title of the video again :o

[@Triantalex]

??..

[@alanzyoutube]

Love the ant walk sound!

[@George-zj9rr]

It's adorable.

[@ShubhamRaj-mu8ol]

Cliff is really the happiest man on Earth. I really believe that just his presence can cheer up even the saddest peson on the planet. I just want to have same passion for a subject as he has for Maths. Really love this guy.

[@GilbertTang]

I'm just in awe. Not only of the concept, but the craftsmanship and raw enthusiasm as well. This is definitely a among the all time great Numberphiles.

[@KeyMan137]

8:45 the most maniacal laughter

[@Tatiana-jt9hd]

*most mathematical maniacal laughter

[@rohansastry]

There once was a person named Klein Who thought Mobius strips were divine And then he said If two combine, You get a cool bottle like mine!

[@bernhardkrickl3567]

Nice, but it bugs me that it started as a limerick but then it wasn't one. How about this: There once was a person named Klein Who thought Mobius strips were divine But if you take two and bond them with glue You get a cool bottle like mine!

[@Mortiis558]

Yeah it needs to be five lines not four.

[@rohansastry]

@@bernhardkrickl3567 Klein invented his bottle in 1882. Glue was invented in 1942.

[@PerMortensen]

@@bernhardkrickl3567 Much better, the meter of the first one bothered me a lot.

[@PerMortensen]

@@rohansastry You must be talking about a specific type of glue. Use of glues and adhesives goes back thousands of years.

[@LokNWykLeer]

Glass: exists Cliff: *math intensifies*

[@Triantalex]

false.

[@EpicMathTime]

The ant klein bottle animation with the different inside/outside crawling sound was so fantastically well done. Don't think that kind of thing goes unnoticed!

[@holypain8152]

I can listen to this man all day, his enthusiasm is just awesome.

[@jeannin2.718]

Cliff is so giddy about these Klein bottles and I am so here for it

[@monoastro]

The glassblower has some mad skills

[@MCLegoboy]

The joy of receiving a Cliff Stoll video is already immeasurable, but to know that there's another one coming up, we're going to have to move into another dimension to calculate that amount of joy!

[@porl42]

Anyone who likes Cliff in these should *definitely* look into ordering a Klein Bottle off him. Even the website is delightfully bonkers haha. I ordered one a while back and it is one of the few things I have insisted on keeping the original box for; everything about the process is amazing (including the pictures of the bottle next to various flowers in his garden as he was preparing to post it). I am in no way affiliated but you all should definitely look at getting one! Just typing this makes me smile thinking of all the little "easter eggs" that came along with ordering it.

[@Laroac]

Those bottles are very klein, indeed.

[@verne8997]

Underrated

[@ryy1704]

I dont get it

[@verne8997]

@@ryy1704 klein translates to small in some languages like German

[@tamirerez2547]

1:50 Special price to the sound effects man.

[@n20games52]

This is very interesting and yet I just kept thinking about the crazy skill of the glass blower. Wow!

[@robsonnascimento356]

I’m not a mathematician and I keep watching all Cliff’s videos just because of his excitement even though I have no idea how a mobius loop or a Klein bottle will ever mean something to me.

[@lukjad007]

I love Cliff's excitement.

[@wallmenis]

Cliff is making me happy! He's such a happy man!

[@Terri_MacKay]

I've never seen anybody get as excited about anything as Cliff gets about his Klein bottles. ❤❤

[@ChickenWolfe]

Thank you so much for my Klein bottle, you have encouraged me and given me so many ideas

[@user-qj7qq6lw2n]

When i grow old, I want to be as happy as this man. Loved him so much.

[@amodgawade4323]

8:46 - evil laugh only a villain with the power of three klein bottles linked internally can do

[@TheTechAdmin]

2:44 I absolutely love this man's enthusiasm regarding breaking topological rules, lol. Like a little kid reacting when you "evenly share something", the funny way. "One for you, one for me". "Two for you, one, two for me". "Three for you, one, two, three for me".

[@TheTechAdmin]

8:45 !!!! I love it!!!!

[@lynk5902]

8:46 The most sinister, yet joyfully exuberant laugh I have ever heard.

[@rossrobbins7707]

His enthusiasm is contagious. Love it!

[@parkerstern1800]

“Well in that case I’ll end up with something...” **maniacal laughing commences**

[@scottarnink9874]

One of my favourite teachers on the planet. He teaches all the while beaming with joy for the lesson. Cliff is a treasure!

[@macronencer]

The skill of these glassblowers BOGGLES my brain. That nested one at the end was fantastic, as well as the 17-loop.

[@TheRealGuywithoutaMustache]

I've heard of Klein Bottles before, it's fascinating how the science behind it works

[@maxhaibara8828]

Oh, you're also interested in Klein Bottles?

[@_vicary]

It’s a miracle that he never broken a Klein bottle with this much excitement every time.

[@RegulatedMilitia]

"Wouldn't it be crazy if i had [weird shaped thing]..." precedes to take out a glass model of that thing

[@crashandsideburns]

As soon as I saw Cliff, I started smiling.

[@jespersmink1467]

Imagine being Cliff’s glasblower: “Allright here are your 3 connected Klein bottles, it was quite a lot of work.” “Hehehe thanks! Now I need 17!”

[@rezandrarizkyirianto-1933]

Glassblower: Say sike right now

[@iiballisticpanda]

This is actually insane, these glass makers must hate requests like this

[@joshuagrosvenor9837]

I love how Cliff laughs like a maniac while unveiling the ultimate Klein bottle.

[@i_am_aladeen]

Cliff is so enthusiastic, he could pass as a nutty professor. I say this with lots of respect :) I love watching his videos.

[@CraftQueenJr]

Aladeen and aladeen he is a nutty professor.

[@The00Chipmunk]

I love Chris Stoll's enthusiasm and passion for what he truly admires. Every time I see him on these he's always so giddy and excited about all that he does and I wish I could take classes from him because I could listen to him all day.

[@julioraitz]

I just love Cliff's videos! He is overly excited with the subjects and it's heartwarming and fun 😍😍😍

[@SwagnerCountsThings]

I love watching Cliff videos because he always gets so excited

[@MCChubbyUnicorn]

I love this guy, he reminds me of doc from back to the future, because of the excitement he gets over everything

[@Dat2Phit]

8:45 me every night

[@Puleczech]

)))

[@bariumselenided5152]

Klein bottle man is my favorite person in history. Dude is just pure joy, and it’s contagious

[@you900001]

I am so excited to see him again! love these videos

[@15october91]

I will call my first child Klein.

[@lilmonarch9935]

Klein is the Dutch word for Little / Small

[@aleksapetrovic7088]

I'll call mine Cliff

[@T-Fyre]

Clif, you’re an absolute mad lad, I LOVE IT

[@JohnLeePettimoreIII]

Cliff's enthusiasm and passion for "his subject" is awe-inspiring. I wish there were more Cliffs in the world.

[@dlevi67]

The snapshot at the end - worth a million (and one) Klein bottles by itself.

[@bobbycraig2583]

cliff's 4D speciality is here to stay!

[@stevedu3316]

Boy I love his enthusiasm

[@thej680]

I love that he has a picture of Fourier in the background. The Fourier Series is a fun way to solve those partial differential equations!

[@christophermclaughlin8899]

No better way to start the day than another Cliff video. Brady, I can’t thank you enough for making these videos, introducing me to the man, the mind, the double-möbius loop, the one and only Cliff Stoll.

[@TerribleTonyShow]

Lucas is a great guy, I'm proud he's this ambitious with his work.

[@Wzymedia]

After all these years, we have finally reached the final boss of Klein bottles

[@silvertheelf]

Him mentioning that the recycling symbol is a mobius loop... me proceeding to mind blown. You can easily tell that the glass blower puts great pride in his work.

[@Nerketur]

Cliff is the most amazing person to watch, listen to, and learn from. He loves his craft, he enjoys discussing it, and show and tell. I'm glad he's doing well, and that 17 Klein bottle ring is amazing! What's next? A 103 Klein bottle Klein bottle?