{"product_id":"klein-bottles-ornaments-transparent-glass-bottle-living-room-wine-cooler-tv-cabinet-desktop-decorative-home-decoration-crafts","title":"Blown glass Klein bottle, mathematical object - TOPOS","description":"\u003cstyle\u003e\n.aa{font-family:Georgia,'Times New Roman',serif;color:#1A1A1A;line-height:1.7;font-size:15px;max-width:760px;margin:0 auto}\n.aa p{margin:0 0 1rem}\n.aa a{color:#185FA5;text-decoration:underline}\n.aa .aa-title{display:inline-block;border-bottom:2px solid #FAC775;padding-bottom:4px;font-size:16px;font-weight:600;margin:1.75rem 0 .5rem;letter-spacing:.3px;color:#1A1A1A}\n.aa .aa-badge{display:inline-block;font-size:11px;font-weight:700;letter-spacing:.5px;text-transform:uppercase;padding:3px 10px;border-radius:20px;margin:0 4px 4px 0;vertical-align:middle;font-family:Georgia,serif}\n.aa .aa-badge-or{background:#FAC775;color:#1A1A1A}\n.aa .aa-badge-brun{background:#5C3A1E;color:#FFF9E8}\n.aa .aa-badge-noir{background:#1A1A1A;color:#FAC775}\n.aa .aa-badge-bleu{background:#185FA5;color:#fff}\n.aa .aa-badge-orange{background:#C84B31;color:#fff}\n.aa .aa-vars{display:grid;grid-template-columns:repeat(2,1fr);gap:12px;margin:1rem 0 1.5rem}\n.aa .aa-var{background:#fff;border:1px solid #d8d2c4;border-radius:8px;overflow:hidden}\n.aa .aa-var-v{height:90px;display:flex;align-items:center;justify-content:center;text-align:center;font-size:13px;font-weight:700;letter-spacing:1px;text-transform:uppercase;padding:0 10px}\n.aa .aa-var-v.dark{color:#FAC775}\n.aa .aa-var-v.light{color:#1A1A1A}\n.aa .aa-var-i{background:#fff;border-top:1px solid #ebe5d6;padding:.6rem .8rem}\n.aa .aa-var-n{font-size:13px;font-weight:700;display:block}\n.aa .aa-var-d{font-size:11px;font-style:italic;color:#555;display:block}\n.aa .aa-mat{background:#fff;border:1px solid #E0DCD3;border-radius:10px;padding:1.2rem 1.4rem;margin:1.5rem 0}\n.aa .aa-mat h3{color:#5C3A1E;font-size:16px;margin:0 0 .6rem}\n.aa .aa-mat p{font-size:14px}\n.aa .aa-ben{display:grid;grid-template-columns:repeat(2,1fr);gap:10px;margin:1rem 0 1.5rem;background:#F5EFE6;padding:12px;border-radius:10px}\n.aa .aa-b{background:#fff;border:1px solid #ebe5d6;border-radius:6px;border-left:3px solid #FAC775;padding:.95rem 1.1rem;display:flex;gap:10px}\n.aa .aa-b .ic{color:#C84B31;font-size:18px;line-height:1.2}\n.aa .aa-b .tt{font-weight:700;font-family:Georgia,serif;font-size:14px;display:block;margin-bottom:2px}\n.aa .aa-b .ds{color:#555;font-size:13px;display:block;line-height:1.55}\n.aa .aa-des{background:linear-gradient(135deg,#1A1A1A,#2A2A2A);border-left:4px solid #FAC775;border-radius:12px;padding:1.4rem 1.6rem;margin:1.5rem 0}\n.aa .aa-des.brun{background:linear-gradient(135deg,#5C3A1E,#704028)}\n.aa .aa-des .t{color:#FAC775;text-transform:uppercase;letter-spacing:.05em;font-size:15px;font-weight:700;margin:0 0 .7rem}\n.aa .aa-des .x{color:#e8e0d8;font-size:13px}\n.aa .aa-des .x p{color:#FFF9E8;font-size:13px}\n.aa .aa-des strong{color:#FAC775}\n.aa .aa-des a{color:#FAC775}\n.aa .aa-steps{background:linear-gradient(135deg,#1A1A1A,#2A2A2A);border-left:4px solid #FAC775;border-radius:6px;padding:1.4rem 1.6rem;margin:1.5rem 0}\n.aa .aa-steps h4{color:#FAC775;text-transform:uppercase;letter-spacing:.05em;font-size:14px;margin:0 0 .8rem}\n.aa .aa-step{display:flex;gap:14px;padding:.75rem 0;border-bottom:1px solid rgba(255,255,255,.1);align-items:flex-start}\n.aa .aa-step:last-child{border-bottom:none}\n.aa .aa-step-n{flex:0 0 32px;width:32px;height:32px;border-radius:50%;background:#FAC775;color:#1A1A1A;font-weight:700;display:flex;align-items:center;justify-content:center;font-size:14px}\n.aa .aa-step-c{color:#e8e0d8;font-size:13px}\n.aa .aa-step-c strong{color:#FAC775}\n.aa .aa-cmp{background:#fff;border:1px solid #ebe5d6;border-radius:6px;padding:1.25rem;margin:1.5rem 0;overflow-x:auto}\n.aa .aa-cmp table,.aa .aa-sp table{width:100%;border-collapse:collapse}\n.aa .aa-cmp th{background:#1A1A1A;color:#FAC775;font-size:11px;text-transform:uppercase;letter-spacing:.5px;padding:.6rem;text-align:left}\n.aa .aa-cmp td{border-bottom:1px solid #ebe5d6;font-size:12px;padding:.55rem .6rem;vertical-align:top}\n.aa .aa-cmp td:first-child{font-weight:700;color:#1A1A1A}\n.aa .aa-cmp tr.hl td{background:#FFF9E8}\n.aa .aa-info,.aa .aa-tip,.aa .aa-warn{border-radius:0 6px 6px 0;padding:1rem 1.2rem;margin:1.25rem 0;font-size:14px}\n.aa .aa-info{background:#E8F4FA;border-left:4px solid #185FA5;color:#0A2A4A}\n.aa .aa-info strong{color:#185FA5}\n.aa .aa-tip{background:#E1F5EE;border-left:4px solid #0F8F6B;color:#04342C}\n.aa .aa-tip strong{color:#0F8F6B}\n.aa .aa-warn{background:#FFF5E8;border-left:4px solid #D89020;color:#5C3A00}\n.aa .aa-warn strong{color:#D89020}\n.aa .aa-info p,.aa .aa-tip p,.aa .aa-warn p{margin:0 0 .5rem}\n.aa .aa-info p:last-child,.aa .aa-tip p:last-child,.aa .aa-warn p:last-child{margin:0}\n.aa .aa-sp{background:#fff;border:1px solid #ebe5d6;border-radius:6px;padding:.4rem 1rem;margin:1rem 0 1.5rem}\n.aa .aa-sp td{padding:.55rem .4rem;border-bottom:1px solid #ebe5d6;font-size:13px;vertical-align:top}\n.aa .aa-sp tr:last-child td{border-bottom:none}\n.aa .aa-sp td:first-child{font-weight:700;color:#1A1A1A;width:45%}\n.aa .aa-gift{background:linear-gradient(135deg,#FFFDF7,#FFF9E8);border:1px solid #FAC775;border-radius:10px;padding:1.3rem 1.5rem;margin:1.5rem 0}\n.aa .aa-gift h3{color:#5C3A1E;font-size:16px;margin:0 0 .6rem}\n.aa .aa-gift p{font-size:14px}\n.aa .aa-faq{margin:1rem 0 1.5rem}\n.aa details.aa-q{border-bottom:1px solid #ebe5d6;padding:.85rem 0}\n.aa details.aa-q summary{font-size:14px;font-weight:700;list-style:none;padding-right:20px;position:relative;cursor:pointer}\n.aa details.aa-q summary::-webkit-details-marker{display:none}\n.aa details.aa-q summary::after{content:\"+\";position:absolute;right:0;top:0;color:#FAC775;font-size:18px;line-height:1}\n.aa details.aa-q[open] summary::after{content:\"\\2212\"}\n.aa .aa-a{font-size:13px;color:#3a3a3a;margin-top:.5rem}\n.aa .aa-a p{margin:0 0 .5rem}\n@media(max-width:600px){.aa .aa-vars,.aa .aa-ben{grid-template-columns:1fr}}\n\u003c\/style\u003e\n\u003cdiv class=\"aa\"\u003e\n\n\u003cp\u003e\u003cspan class=\"aa-badge aa-badge-noir\"\u003eThe Lab\u003c\/span\u003e \u003cspan class=\"aa-badge aa-badge-bleu\"\u003eMathematics\u003c\/span\u003e Here is one of the most fascinating objects in geometry: the \u003cstrong\u003eKlein bottle\u003c\/strong\u003e, a closed surface that has \u003cstrong\u003eneither an inside nor an outside\u003c\/strong\u003e. Our TOPOS model provides a tangible interpretation of it in \u003cstrong\u003ehand-blown glass\u003c\/strong\u003e, perfectly transparent, allowing you to follow with your eyes the path of the neck diving through the wall to join the bottom. Offered in \u003cstrong\u003etwo sizes\u003c\/strong\u003e — a small one, approximately 10 × 6 cm, and a large one, approximately 16 × 9 cm — it sits on a desk, a bookshelf, or in a display case as a true \u003cstrong\u003emathematical sculpture\u003c\/strong\u003e. Conceived by the mathematician \u003cstrong\u003eFelix Klein\u003c\/strong\u003e in 1882, this form is a cousin of the Möbius strip and intrigues both scientific minds and design enthusiasts. Each piece, shaped by a \u003cstrong\u003eglassblower\u003c\/strong\u003e, presents slight variations that make it a unique object. Sold empty, it can remain as it is, crystalline and graphic, or serve as a starting point for a demonstration with a colored liquid. A rare \u003cstrong\u003eobject of curiosity\u003c\/strong\u003e that makes topology speak for itself.\u003c\/p\u003e\n\n\u003ch3 class=\"aa-title\"\u003eThe two available sizes\u003c\/h3\u003e\n\n\u003cdiv class=\"aa-vars\"\u003e\n\n\u003cdiv class=\"aa-var\"\u003e\n\n\u003cdiv class=\"aa-var-v light\" style=\"background: linear-gradient(135deg,#F5F0E8,#E8F4FA);\"\u003eSMALL · 10 × 6 CM\u003c\/div\u003e\n\n\u003cdiv class=\"aa-var-i\"\u003e\n\n\u003cspan class=\"aa-var-n\"\u003eSmall – 10 × 6 cm\u003c\/span\u003e\u003cspan class=\"aa-var-d\"\u003eCompact size for a desk, a shelf, or a collector's display case\u003c\/span\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-var\"\u003e\n\n\u003cdiv class=\"aa-var-v light\" style=\"background: linear-gradient(135deg,#E8F4FA,#F5F0E8);\"\u003eLARGE · 16 × 9 CM\u003c\/div\u003e\n\n\u003cdiv class=\"aa-var-i\"\u003e\n\n\u003cspan class=\"aa-var-n\"\u003eLarge – 16 × 9 cm\u003c\/span\u003e\u003cspan class=\"aa-var-d\"\u003ePresentation size, approximately 17 cm high and 9.5 cm wide based on photo measurements\u003c\/span\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cp\u003eBoth sizes feature the same shape and the same transparent glass. The small one slides in anywhere and makes a lovely desk object; the large one offers a more assertive presence and easier visualization of the neck passing through the wall, ideal for a demonstration or showcasing on a bookshelf.\u003c\/p\u003e\n\n\u003cdiv class=\"aa-mat\"\u003e\n\n\u003ch3\u003eThe Klein bottle: a surface without edges or sides\u003c\/h3\u003e\n\n\u003cp\u003eIn \u003cstrong\u003etopology\u003c\/strong\u003e, the branch of mathematics that studies shapes independently of their measurements, the Klein bottle is a \u003cstrong\u003enon-orientable surface\u003c\/strong\u003e: it is impossible to distinguish an \"inside\" from an \"outside.\" An ant walking along its wall could move from what seems to be the inner face to the outer face without ever crossing an edge. Described by Felix Klein in 1882, it is a close relative of the \u003cstrong\u003eMöbius strip\u003c\/strong\u003e: by gluing two Möbius strips together along their edges, you obtain exactly a Klein bottle.\u003c\/p\u003e\n\n\u003cp\u003eIn our three-dimensional space, a true Klein bottle cannot exist without \u003cstrong\u003epassing through itself\u003c\/strong\u003e: this is why the glass neck dives through the wall before joining the bottom. Mathematicians call this an \u003cstrong\u003eimmersion\u003c\/strong\u003e; in a four-dimensional space, the surface could close up without any intersection. \u003cstrong\u003eBlown glass\u003c\/strong\u003e, a transparent and continuous material, is ideal for making this continuity visible: you can follow the transition from one wall to the other with your eyes.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003ch3 class=\"aa-title\"\u003eWhy you will love it\u003c\/h3\u003e\n\n\u003cdiv class=\"aa-ben\"\u003e\n\n\u003cdiv class=\"aa-b\"\u003e\n\n\u003cspan class=\"ic\"\u003e✦\u003c\/span\u003e\n\u003cdiv\u003e\n\n\u003cspan class=\"tt\"\u003eA mathematical icon\u003c\/span\u003e\u003cspan class=\"ds\"\u003eA concept studied in topology for over a century, transformed into an object you can hold and observe.\u003c\/span\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-b\"\u003e\n\n\u003cspan class=\"ic\"\u003e◆\u003c\/span\u003e\n\u003cdiv\u003e\n\n\u003cspan class=\"tt\"\u003eHand-blown glass\u003c\/span\u003e\u003cspan class=\"ds\"\u003eEach piece is shaped by a glassblower: slight variations make each bottle a unique specimen.\u003c\/span\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-b\"\u003e\n\n\u003cspan class=\"ic\"\u003e➤\u003c\/span\u003e\n\u003cdiv\u003e\n\n\u003cspan class=\"tt\"\u003eTotal transparency\u003c\/span\u003e\u003cspan class=\"ds\"\u003eThe clear glass lets you follow the path of the neck through the wall, making the form understandable at a glance.\u003c\/span\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-b\"\u003e\n\n\u003cspan class=\"ic\"\u003e✦\u003c\/span\u003e\n\u003cdiv\u003e\n\n\u003cspan class=\"tt\"\u003eTwo formats\u003c\/span\u003e\u003cspan class=\"ds\"\u003eSmall for a desk or display case, large for an assertive presentation or demonstration.\u003c\/span\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-b\"\u003e\n\n\u003cspan class=\"ic\"\u003e◆\u003c\/span\u003e\n\u003cdiv\u003e\n\n\u003cspan class=\"tt\"\u003eAn educational tool\u003c\/span\u003e\u003cspan class=\"ds\"\u003eIdeal for explaining notions of orientability, closed surfaces, and dimensions to a curious audience.\u003c\/span\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-b\"\u003e\n\n\u003cspan class=\"ic\"\u003e➤\u003c\/span\u003e\n\u003cdiv\u003e\n\n\u003cspan class=\"tt\"\u003eA conversation piece\u003c\/span\u003e\u003cspan class=\"ds\"\u003ePlaced on a desk, it inevitably intrigues and opens discussions about geometry and infinity.\u003c\/span\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-des\"\u003e\n\n\u003cp class=\"t\"\u003eThe paradox under glass\u003c\/p\u003e\n\n\u003cdiv class=\"x\"\u003e\n\n\u003cp\u003eThere are objects that you understand by looking at them, and others that you look at to try and understand them. The Klein bottle belongs to the second category. The eye follows the \u003cstrong\u003estretching neck\u003c\/strong\u003e, which curves, passes through the wall, and then opens at the bottom to join the outer surface. Where does the inside begin? Where does the outside end? The question has no answer, and that is the charm of the shape.\u003c\/p\u003e\n\n\u003cp\u003eThe \u003cstrong\u003eblown glass\u003c\/strong\u003e gives this paradox an almost organic elegance. Light slides over the curves, diffracts through the thickness, and creates reflections that change according to the time of day. Empty, the bottle is a \u003cstrong\u003ecrystalline sculpture\u003c\/strong\u003e; filled with a bit of colored water for the duration of a demonstration, it suddenly reveals the continuity of its surface.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-steps\"\u003e\n\n\u003ch4\u003eShowcasing your Klein bottle\u003c\/h4\u003e\n\n\u003cdiv class=\"aa-step\"\u003e\n\n\u003cspan class=\"aa-step-n\"\u003e1\u003c\/span\u003e\n\u003cdiv class=\"aa-step-c\"\u003e\n\n\u003cstrong\u003eUnpack it with care\u003c\/strong\u003e, holding it by the main body, never by the thin neck, which is the most delicate part of the blown glass.\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-step\"\u003e\n\n\u003cspan class=\"aa-step-n\"\u003e2\u003c\/span\u003e\n\u003cdiv class=\"aa-step-c\"\u003e\n\n\u003cstrong\u003eChoose a stable location\u003c\/strong\u003e: a bookshelf, desk, or display case, away from high-traffic areas and the edges of furniture.\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-step\"\u003e\n\n\u003cspan class=\"aa-step-n\"\u003e3\u003c\/span\u003e\n\u003cdiv class=\"aa-step-c\"\u003e\n\n\u003cstrong\u003ePlay with light\u003c\/strong\u003e: placed near a window or a lamp, the bottle captures reflections that highlight its curves.\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-step\"\u003e\n\n\u003cspan class=\"aa-step-n\"\u003e4\u003c\/span\u003e\n\u003cdiv class=\"aa-step-c\"\u003e\n\n\u003cstrong\u003ePerform the demonstration\u003c\/strong\u003e if you wish: pour a little colored water using a thin funnel to visualize the continuity of the surface, then empty and dry.\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\n\u003ch3 class=\"aa-title\"\u003eKlein bottle or Möbius strip?\u003c\/h3\u003e\n\n\u003cdiv class=\"aa-cmp\"\u003e\n\n\u003ctable\u003e\n\n\u003cthead\u003e\n\n\u003ctr\u003e\n\n\u003cth\u003eCriteria\u003c\/th\u003e\n\n\u003cth\u003eSmall Klein bottle\u003c\/th\u003e\n\n\u003cth\u003eLarge Klein bottle\u003c\/th\u003e\n\n\u003cth\u003eMöbius strip\u003c\/th\u003e\n\n\n\u003c\/tr\u003e\n\n\n\u003c\/thead\u003e\n\n\u003ctbody\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eNature\u003c\/td\u003e\n\n\u003ctd\u003eNon-orientable closed surface\u003c\/td\u003e\n\n\u003ctd\u003eNon-orientable closed surface\u003c\/td\u003e\n\n\u003ctd\u003eNon-orientable surface with one edge\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr class=\"hl\"\u003e\n\n\u003ctd\u003eEdge\u003c\/td\u003e\n\n\u003ctd\u003eNone\u003c\/td\u003e\n\n\u003ctd\u003eNone\u003c\/td\u003e\n\n\u003ctd\u003eA single edge\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eFormat\u003c\/td\u003e\n\n\u003ctd\u003eApprox. 10 × 6 cm\u003c\/td\u003e\n\n\u003ctd\u003eApprox. 16 × 9 cm\u003c\/td\u003e\n\n\u003ctd\u003eVariable (paper strip)\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr class=\"hl\"\u003e\n\n\u003ctd\u003eMaterial\u003c\/td\u003e\n\n\u003ctd\u003eTransparent blown glass\u003c\/td\u003e\n\n\u003ctd\u003eTransparent blown glass\u003c\/td\u003e\n\n\u003ctd\u003ePaper, metal, or other\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eUsage\u003c\/td\u003e\n\n\u003ctd\u003eDesk object, display case\u003c\/td\u003e\n\n\u003ctd\u003ePresentation piece, demonstration\u003c\/td\u003e\n\n\u003ctd\u003eSimple DIY experiment\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\n\u003c\/tbody\u003e\n\n\n\u003c\/table\u003e\n\n\n\u003c\/div\u003e\n\n\u003cp\u003eTo continue exploring shapes and enigmas, discover our selection of \u003ca href=\"\/collections\/casse-tetes-et-bureau\"\u003epuzzles and desk objects\u003c\/a\u003e, where logical challenges meet objects of curiosity.\u003c\/p\u003e\n\n\u003cdiv class=\"aa-info\"\u003e\n\n\u003cp\u003e\u003cstrong\u003eGood to know:\u003c\/strong\u003e the bottle is \u003cstrong\u003esold empty\u003c\/strong\u003e. The amber or blue liquids visible in some photos are used solely for staging and demonstration purposes. Manual blowing means that the shape, glass thickness, and dimensions may vary slightly from one piece to another: this is the signature of artisanal work, not a defect.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-tip\"\u003e\n\n\u003cp\u003e\u003cstrong\u003eTip:\u003c\/strong\u003e for a spectacular demonstration, use water mixed with a drop of food coloring and a \u003cstrong\u003ethin funnel\u003c\/strong\u003e. Gently tilt the bottle: the liquid passes from the \"outer wall\" to the \"inner wall\" without crossing any edge. Then empty completely and let air dry, opening facing down.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003ch3 class=\"aa-title\"\u003eTechnical specifications\u003c\/h3\u003e\n\n\u003cdiv class=\"aa-sp\"\u003e\n\n\u003ctable\u003e\n\n\u003ctbody\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eType\u003c\/td\u003e\n\n\u003ctd\u003eDecorative and educational Klein bottle\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eMaterial\u003c\/td\u003e\n\n\u003ctd\u003eTransparent glass\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eManufacturing\u003c\/td\u003e\n\n\u003ctd\u003eManual blowing\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eSmall size\u003c\/td\u003e\n\n\u003ctd\u003eApprox. 10 × 6 cm\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eLarge size\u003c\/td\u003e\n\n\u003ctd\u003eApprox. 16 × 9 cm (photo measurement: 17 cm high, 9.5 cm wide)\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eDelivery contents\u003c\/td\u003e\n\n\u003ctd\u003eSold empty\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eField\u003c\/td\u003e\n\n\u003ctd\u003eMathematics, topology\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eRecommended age\u003c\/td\u003e\n\n\u003ctd\u003e14 years and older\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eWeight\u003c\/td\u003e\n\n\u003ctd\u003eNot disclosed\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eUsage\u003c\/td\u003e\n\n\u003ctd\u003eDecoration, desk object, educational demonstration\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\u003ctr\u003e\n\n\u003ctd\u003eCare\u003c\/td\u003e\n\n\u003ctd\u003eRinse with clear water, air dry\u003c\/td\u003e\n\n\n\u003c\/tr\u003e\n\n\n\u003c\/tbody\u003e\n\n\n\u003c\/table\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-warn\"\u003e\n\n\u003cp\u003e\u003cstrong\u003ePrecautions:\u003c\/strong\u003e \u003cstrong\u003efragile blown glass\u003c\/strong\u003e object, handle with care, avoiding grasping it by the neck. Do not expose to thermal shocks (boiling water, sudden changes from cold to hot), do not put in the dishwasher. Recommended from \u003cstrong\u003e14 years old\u003c\/strong\u003e; keep out of reach of young children. Do not use as a food container or for drinking. In case of breakage, clean up fragments with caution.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003cdiv class=\"aa-gift\"\u003e\n\n\u003ch3\u003eA gift idea for curious minds\u003c\/h3\u003e\n\n\u003cp\u003eTOPOS is the \u003cstrong\u003eideal gift for a mathematician\u003c\/strong\u003e, teacher, science student, or engineer: a learned nod that will find its place on a desk or in a library. It also marks a \u003cstrong\u003egraduation\u003c\/strong\u003e, retirement, or birthday with a touch of originality.\u003c\/p\u003e\n\n\u003cp\u003eTo assemble a science-themed gift, browse \u003ca href=\"\/collections\/objets-scientifiques\"\u003eThe Lab's scientific objects\u003c\/a\u003e and pair it with another object of curiosity.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003ch3 class=\"aa-title\"\u003eFrequently asked questions\u003c\/h3\u003e\n\n\u003cdiv class=\"aa-faq\"\u003e\n\n\u003cdetails class=\"aa-q\"\u003e\n\u003csummary\u003eWhat is a Klein bottle?\u003c\/summary\u003e\n\u003cdiv class=\"aa-a\"\u003e\n\n\u003cp\u003eThe Klein bottle is a \u003cstrong\u003enon-orientable closed surface\u003c\/strong\u003e described by the German mathematician \u003cstrong\u003eFelix Klein\u003c\/strong\u003e in 1882. Unlike an ordinary bottle, it has no distinct inside or outside: by following its wall, you pass from one face to the other without ever encountering an edge. It is closely linked to the \u003cstrong\u003eMöbius strip\u003c\/strong\u003e: by assembling two Möbius strips along their edges, you obtain a Klein bottle. In our three-dimensional space, it can only be represented by passing through itself, hence the neck that plunges through the wall. It is one of the iconic objects of topology, often used to illustrate the concept of orientability.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003c\/details\u003e\n\u003cdetails class=\"aa-q\"\u003e\n\u003csummary\u003eWhy does the neck go through the wall?\u003c\/summary\u003e\n\u003cdiv class=\"aa-a\"\u003e\n\n\u003cp\u003eA true Klein bottle has \u003cstrong\u003eno intersection\u003c\/strong\u003e with itself, but this is only possible in a \u003cstrong\u003efour-dimensional\u003c\/strong\u003e space. In our three-dimensional world, the only way to physically represent it is to pass the neck through the wall: mathematicians speak of \u003cstrong\u003eimmersion\u003c\/strong\u003e rather than embedding. The passing of the neck through the glass is therefore a geometric constraint, not an aesthetic choice. It is precisely this detail that makes the object so intriguing: it concretely shows the limits of our space and invites us to imagine an additional dimension in which the surface would perfectly close back on itself.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003c\/details\u003e\n\u003cdetails class=\"aa-q\"\u003e\n\u003csummary\u003eWhich size should I choose?\u003c\/summary\u003e\n\u003cdiv class=\"aa-a\"\u003e\n\n\u003cp\u003eThe \u003cstrong\u003esmall\u003c\/strong\u003e (approx. 10 × 6 cm) is ideal for a desk, an already crowded shelf, or a collector's display case: discreet, it intrigues without cluttering. The \u003cstrong\u003elarge\u003c\/strong\u003e (approx. 16 × 9 cm, nearly 17 cm high based on photo measurements) offers a more assertive presence and better legibility of the neck's path through the wall, making it an excellent \u003cstrong\u003edemonstration support\u003c\/strong\u003e in a classroom or during a conference. For a gift for a mathematics enthusiast, the large size often has more impact; for an everyday desk object, the small is perfect. Dimensions may vary slightly as each piece is hand-blown.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003c\/details\u003e\n\u003cdetails class=\"aa-q\"\u003e\n\u003csummary\u003eIs the bottle delivered filled with liquid?\u003c\/summary\u003e\n\u003cdiv class=\"aa-a\"\u003e\n\n\u003cp\u003eNo, the Klein bottle is \u003cstrong\u003esold empty\u003c\/strong\u003e. The amber or blue liquids seen in some photos are used solely for \u003cstrong\u003estaging\u003c\/strong\u003e and to illustrate the principle of surface continuity. You can certainly keep it empty, like a transparent glass sculpture, or fill it occasionally with colored water for a demonstration. In that case, use a thin funnel, avoid staining, greasy, or corrosive liquids, then empty it and let it \u003cstrong\u003eair dry\u003c\/strong\u003e, opening turned downwards. Complete drying may take some time due to the shape of the neck.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003c\/details\u003e\n\u003cdetails class=\"aa-q\"\u003e\n\u003csummary\u003eIs every bottle identical?\u003c\/summary\u003e\n\u003cdiv class=\"aa-a\"\u003e\n\n\u003cp\u003eNot exactly, and that is intentional. Each Klein bottle is \u003cstrong\u003ehand-blown\u003c\/strong\u003e by a glass artisan: the curvature of the neck, the thickness of the glass, the position of the passage through the wall, and the exact dimensions may vary slightly from one piece to another. Small \u003cstrong\u003eair bubbles\u003c\/strong\u003e or fine irregularities may also appear in the glass: these testify to manual work and are not defects. These variations make each specimen a \u003cstrong\u003eunique piece\u003c\/strong\u003e, which adds to the character of the object, especially when offered as a gift. The general shape and the topological principle naturally remain identical.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003c\/details\u003e\n\u003cdetails class=\"aa-q\"\u003e\n\u003csummary\u003eHow do I clean the Klein bottle?\u003c\/summary\u003e\n\u003cdiv class=\"aa-a\"\u003e\n\n\u003cp\u003eFor the exterior, a \u003cstrong\u003esoft, dry cloth\u003c\/strong\u003e or a slightly damp one is sufficient. For the interior, if the bottle has contained liquid, rinse it several times with \u003cstrong\u003elukewarm water\u003c\/strong\u003e (never boiling) by circulating the water through the opening, then empty it completely by tilting it in several directions. Then let it \u003cstrong\u003eair dry\u003c\/strong\u003e, opening facing down, placed on a stable surface. Do not use stiff brushes, abrasive products, or a dishwasher: blown glass is thin and sensitive to thermal shock. If limescale marks appear, rinsing with demineralized water usually prevents them.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003c\/details\u003e\n\u003cdetails class=\"aa-q\"\u003e\n\u003csummary\u003eIs it a good gift for a child?\u003c\/summary\u003e\n\u003cdiv class=\"aa-a\"\u003e\n\n\u003cp\u003eThe Klein bottle is recommended for ages \u003cstrong\u003e14 and up\u003c\/strong\u003e. It is a \u003cstrong\u003efragile blown glass\u003c\/strong\u003e object that requires careful handling, especially around the neck. For a science-loving teenager, however, it is a stimulating gift that opens the door to \u003cstrong\u003etopology\u003c\/strong\u003e and mathematics beyond the school curriculum. For a younger child, it is better to present it under adult supervision, placed out of reach, as an object to observe rather than to handle. Paired with a paper Möbius strip, it allows for a very clear little discovery session.\u003c\/p\u003e\n\n\n\u003c\/div\u003e\n\n\u003c\/details\u003e\n\n\u003c\/div\u003e\n\n\n\u003c\/div\u003e\n\u003cp\u003e \u003c\/p\u003e\n\u003cscript type=\"application\/ld+json\"\u003e{\"@context\": \"https:\/\/schema.org\", \"@type\": \"FAQPage\", \"mainEntity\": [{\"@type\": \"Question\", \"name\": \"Qu’est-ce qu’une bouteille de Klein ?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"C’est une surface fermée non orientable décrite par Felix Klein en 1882. Elle n’a ni intérieur ni extérieur distincts : en suivant sa paroi, on passe d’une face à l’autre sans rencontrer de bord. Assembler deux rubans de Möbius par leur bord donne une bouteille de Klein. En trois dimensions, elle ne peut exister qu’en se traversant elle-même, d’où le col qui plonge à travers la paroi.\"}}, {\"@type\": \"Question\", \"name\": \"Pourquoi le col traverse-t-il la paroi ?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"Une vraie bouteille de Klein ne se recoupe pas, mais cela n’est possible que dans un espace à quatre dimensions. En trois dimensions, la seule façon de la représenter est de faire passer le col à travers la paroi : on parle d’immersion. Ce passage est donc une contrainte géométrique, et c’est précisément ce détail qui rend l’objet si intrigant et pédagogique.\"}}, {\"@type\": \"Question\", \"name\": \"Quelle taille choisir ?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"La petite, environ 10 × 6 cm, convient à un bureau, une étagère ou une vitrine. La grande, environ 16 × 9 cm, soit près de 17 cm de haut d’après la cote photo, a plus de présence et rend le trajet du col plus lisible, idéale pour une démonstration ou un cadeau marquant. Les dimensions peuvent légèrement varier, chaque pièce étant soufflée à la main.\"}}, {\"@type\": \"Question\", \"name\": \"La bouteille est-elle livrée remplie de liquide ?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"Non, elle est vendue vide. Les liquides ambrés ou bleus visibles sur certaines photos servent uniquement à la mise en scène. Vous pouvez la garder vide, comme une sculpture de verre, ou la remplir ponctuellement d’eau colorée à l’aide d’un entonnoir fin. Videz-la ensuite et laissez-la sécher à l’air, ouverture vers le bas ; évitez les liquides gras, tachants ou corrosifs.\"}}, {\"@type\": \"Question\", \"name\": \"Chaque bouteille est-elle identique ?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"Pas tout à fait. Chaque bouteille est soufflée à la main : courbure du col, épaisseur du verre et dimensions exactes peuvent varier légèrement. De petites bulles d’air ou irrégularités peuvent apparaître dans le verre ; elles témoignent du travail manuel et ne sont pas des défauts. Chaque exemplaire est ainsi singulier, tandis que la forme générale et le principe topologique restent identiques.\"}}, {\"@type\": \"Question\", \"name\": \"Comment nettoyer la bouteille de Klein ?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"Pour l’extérieur, un chiffon doux suffit. Si la bouteille a contenu un liquide, rincez-la à l’eau tiède, jamais bouillante, en faisant circuler l’eau par l’ouverture, puis videz-la en l’inclinant dans plusieurs directions. Laissez sécher à l’air, ouverture vers le bas. Évitez brosse rigide, produit abrasif et lave-vaisselle ; l’eau déminéralisée limite les traces de calcaire.\"}}, {\"@type\": \"Question\", \"name\": \"Est-ce un bon cadeau pour un enfant ?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"Elle est recommandée à partir de 14 ans. C’est un objet en verre soufflé fragile, qui demande une manipulation attentive, surtout au niveau du col. Pour un adolescent passionné de sciences, c’est un cadeau stimulant qui ouvre la porte à la topologie. Pour un plus jeune, présentez-la sous la surveillance d’un adulte, posée hors de portée, comme un objet à observer.\"}}]}\u003c\/script\u003e\n","brand":"Atelier Atypique","offers":[{"title":"Small","offer_id":54860809404739,"sku":"1005008886771644-10x6cm","price":50.0,"currency_code":"USD","in_stock":true},{"title":"Great","offer_id":54860809437507,"sku":"1005008886771644-16x9cm","price":65.0,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0231\/3365\/0991\/files\/bouteille-de-klein-2.jpg?v=1791060504","url":"https:\/\/atelier-atypique.com\/products\/klein-bottles-ornaments-transparent-glass-bottle-living-room-wine-cooler-tv-cabinet-desktop-decorative-home-decoration-crafts","provider":"Atelier Atypique","version":"1.0","type":"link"}