Gauss–Lucas theorem: Difference between revisions
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{| class="wikitable" align="right" style="margin-left:10px" width="320" | |||
!bgcolor=#e7dcc3 colspan=2|16-cell honeycomb | |||
|- | |||
|bgcolor=#ffffff align=center colspan=2|[[Image:Demitesseractic tetra hc.png|280px]]<br>[[Perspective projection]]: the first layer of adjacent 16-cell facets. | |||
|- | |||
|bgcolor=#e7dcc3|Type||[[List_of_regular_polytopes#Five_Dimensions_2|Regular 4-space honeycomb]]<BR>[[Uniform_polyteron#Regular_and_uniform_honeycombs|Uniform 4-honeycomb]] | |||
|- | |||
|bgcolor=#e7dcc3|Family||[[Alternated hypercube honeycomb]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{3,3,4,3} | |||
|- | |||
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|3|node|3|node|4|node|3|node}} | |||
|- | |||
|bgcolor=#e7dcc3|4-face type||[[16-cell|{3,3,4}]] [[File:Schlegel_wireframe_16-cell.png|40px]] | |||
|- | |||
|bgcolor=#e7dcc3|Cell type||[[tetrahedron|{3,3}]] [[File:Tetrahedron.png|20px]] | |||
|- | |||
|bgcolor=#e7dcc3|Face type||[[Triangle|{3}]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Edge figure]]||[[cube]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:24-cell_t0_F4.svg|80px]]<BR>[[24-cell]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{F}}_4</math> = [3,3,4,3] | |||
|- | |||
|bgcolor=#e7dcc3|Dual||[[Icositetrachoric honeycomb|{3,4,3,3}]] | |||
|- | |||
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]], [[edge-transitive]], [[face-transitive]], [[cell-transitive]] | |||
|} | |||
In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''16-cell honeycomb''' is the one of three regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 4-space. The other two are the [[tesseractic honeycomb]] and the [[24-cell honeycomb]]. This honeycomb is constructed from [[16-cell]] [[Facet (mathematics)|facet]]s, three around every edge. It has a [[24-cell]] [[vertex figure]]. | |||
This [[vertex arrangement]] or lattice is called the B<sub>4</sub>, D<sub>4</sub>, or [[F4 (mathematics)#F4_lattice|F<sub>4</sub> lattice]].<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html</ref><ref name="www2.research.att.com">http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html</ref> | |||
== Alternate names== | |||
* Hexadecachoric tetracomb/honeycomb | |||
* Demitesseractic tetracomb/honeycomb | |||
==Coordinates== | |||
As a regular honeycomb, {3,3,4,3}, it has no lower dimensional analogues, but as an [[Alternation (geometry)|alternated]] form (the '''demitesseractic honeycomb''', h{4,3,3,4}) it is related to the [[alternated cubic honeycomb]]. | |||
Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even. | |||
== D4 lattice == | |||
Its [[vertex arrangement]] is called the [[D4 lattice|D<sub>4</sub> lattice]] or ''F<sub>4</sub> lattice''.<ref name="www2.research.att.com"/> The vertices of this lattice are the centers of the [[3-sphere]]s in the densest possible [[sphere packing|packing]] of equal spheres in 4-space; its [[kissing number]] is 24, which is also the highest possible in 4-space.<ref name="Musin">{{cite journal |author=O. R. Musin |title=The problem of the twenty-five spheres |year=2003 |journal=Russ. Math. Surv. |volume=58 |pages=794–795 |doi=10.1070/RM2003v058n04ABEH000651}}</ref> | |||
:{{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_1|3|node|3|node|4|node|3|node}} | |||
The D{{sup sub|+|4}} lattice (also called D{{sup sub|2|4}}) can be constructed by the union of two 4-demicubic lattices, and is identical to the [[tesseractic honeycomb]]: | |||
:{{CDD|nodes_10ru|split2|node|split1|nodes}} + {{CDD|nodes_01rd|split2|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|3|node|4|node}} | |||
This packing is only a lattice for even dimensions. The kissing number is 2<sup>3</sup>=8, (2<sup>n-1</sup> for n<8, 240 for n=8, and 2n(n-1) for n>8).<ref>Conway (1998), p. 119</ref> | |||
The D{{sup sub|*|4}} lattice (also called D{{sup sub|4|4}} and C{{sup sub|2|4}}) can be constructed by the union of all four 5-demicubic honeycombs, but it is identical to the ''D<sub>4</sub> lattice'': It is also the 4-dimensional [[body centered cubic]], the union of two [[4-cube honeycomb]]s in dual positions. | |||
:{{CDD|nodes_10ru|split2|node|split1|nodes}} + {{CDD|nodes_01rd|split2|node|split1|nodes}} + {{CDD|nodes|split2|node|split1|nodes_10lu}} + {{CDD|nodes|split2|node|split1|nodes_01ld}} = {{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|3|node|4|node}} + {{CDD|node|4|node|3|node|3|node|4|node_1}}. | |||
The [[kissing number]] of the D{{sup sub|*|4}} lattice (and D<sub>4</sub> lattice) is 24<ref>Conway (1998), p. 120</ref> and its [[Voronoi tessellation]] is a [[24-cell honeycomb]], {{CDD|node_1|split1|nodes|4a4b|nodes}}, containing all rectified 16-cells ([[24-cell]]) [[Voronoi cell]]s, {{CDD|node|4|node|3|node_1|3|node}} or {{CDD|node_1|3|node|4|node|3|node}}.<ref>Conway (1998), p. 466</ref> | |||
== Symmetry constructions == | |||
There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored [[16-cell]] facets. | |||
{|class='wikitable' | |||
!Name | |||
![[Coxeter group]] | |||
![[Schläfli symbol]] | |||
![[Coxeter diagram]] | |||
![[Vertex figure]]<BR>Symmetry | |||
![[Facet (geometry)|Facets]]/verf | |||
|- | |||
!16-cell honeycomb | |||
|<math>{\tilde{F}}_4</math> = [3,3,4,3]||{3,3,4,3}||{{CDD|node_1|3|node|3|node|4|node|3|node}}||{{CDD|node_1|3|node|4|node|3|node}}<BR>[3,4,3], order 1152||24: [[16-cell]] | |||
|- | |||
!4-demicube honeycomb | |||
|<math>{\tilde{B}}_4</math> = [3<sup>1,1</sup>,3,4]||{3<sup>1,1</sup>,3,4}<BR> = h{4,3,3,4}||{{CDD|nodes_10ru|split2|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|4|node}}||{{CDD|node|3|node_1|3|node|4|node}}<BR>[3,3,4], order 384||16+8: [[16-cell]] | |||
|- | |||
! | |||
|<math>{\tilde{D}}_4</math> = [3<sup>1,1,1,1</sup>]||{3<sup>1,1,1,1</sup>}<BR> = h{4,3,3<sup>1,1</sup>}||{{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|split1|nodes}}||{{CDD|node|3|node_1|split1|nodes}}<BR>[3<sup>1,1,1</sup>], order 192||8+8+8: [[16-cell]] | |||
|} | |||
== Related honeycombs== | |||
{{F4 honeycombs}} | |||
{{C4 honeycombs}} | |||
{{B4 honeycombs}} | |||
{{D4 honeycombs}} | |||
== See also == | |||
Regular and uniform honeycombs in 4-space: | |||
*[[Tesseractic honeycomb]] | |||
*[[24-cell honeycomb]] | |||
*[[Truncated 24-cell honeycomb]] | |||
*[[Snub 24-cell honeycomb]] | |||
* [[5-cell honeycomb]] | |||
* [[Truncated 5-cell honeycomb]] | |||
* [[Omnitruncated 5-cell honeycomb]] | |||
==Notes== | |||
{{reflist}} | |||
== References == | |||
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 | |||
** pp. 154–156: Partial truncation or alternation, represented by ''h'' prefix: h{4,4} = {4,4}; h{4,3,4} = {3<sup>1,1</sup>,4}, h{4,3,3,4} = {3,3,4,3}, ... | |||
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] | |||
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] | |||
* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' | |||
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} x3o3o4o3o - hext - O104 | |||
* {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}} | |||
{{Honeycombs}} | |||
{{DEFAULTSORT:Demitesseractic Honeycomb}} | |||
[[Category:Honeycombs (geometry)]] | |||
[[Category:5-polytopes]] | |||
Revision as of 23:21, 23 December 2013
| 16-cell honeycomb | |
|---|---|
 Perspective projection: the first layer of adjacent 16-cell facets. | |
| Type | Regular 4-space honeycomb Uniform 4-honeycomb |
| Family | Alternated hypercube honeycomb |
| Schläfli symbol | {3,3,4,3} |
| Coxeter-Dynkin diagram | Template:CDD |
| 4-face type | {3,3,4} 
|
| Cell type | {3,3} 
|
| Face type | {3} |
| Edge figure | cube |
| Vertex figure |  24-cell |
| Coxeter group | = [3,3,4,3] |
| Dual | {3,4,3,3} |
| Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive |
In four-dimensional Euclidean geometry, the 16-cell honeycomb is the one of three regular space-filling tessellation (or honeycomb) in Euclidean 4-space. The other two are the tesseractic honeycomb and the 24-cell honeycomb. This honeycomb is constructed from 16-cell facets, three around every edge. It has a 24-cell vertex figure.
This vertex arrangement or lattice is called the B4, D4, or F4 lattice.[1][2]
Alternate names
- Hexadecachoric tetracomb/honeycomb
- Demitesseractic tetracomb/honeycomb
Coordinates
As a regular honeycomb, {3,3,4,3}, it has no lower dimensional analogues, but as an alternated form (the demitesseractic honeycomb, h{4,3,3,4}) it is related to the alternated cubic honeycomb.
Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.
D4 lattice
Its vertex arrangement is called the D4 lattice or F4 lattice.[2] The vertices of this lattice are the centers of the 3-spheres in the densest possible packing of equal spheres in 4-space; its kissing number is 24, which is also the highest possible in 4-space.[3]
The DTemplate:Sup sub lattice (also called DTemplate:Sup sub) can be constructed by the union of two 4-demicubic lattices, and is identical to the tesseractic honeycomb:
This packing is only a lattice for even dimensions. The kissing number is 23=8, (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[4]
The DTemplate:Sup sub lattice (also called DTemplate:Sup sub and CTemplate:Sup sub) can be constructed by the union of all four 5-demicubic honeycombs, but it is identical to the D4 lattice: It is also the 4-dimensional body centered cubic, the union of two 4-cube honeycombs in dual positions.
- Template:CDD + Template:CDD + Template:CDD + Template:CDD = Template:CDD = Template:CDD + Template:CDD.
The kissing number of the DTemplate:Sup sub lattice (and D4 lattice) is 24[5] and its Voronoi tessellation is a 24-cell honeycomb, Template:CDD, containing all rectified 16-cells (24-cell) Voronoi cells, Template:CDD or Template:CDD.[6]
Symmetry constructions
There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 16-cell facets.
| Name | Coxeter group | Schläfli symbol | Coxeter diagram | Vertex figure Symmetry |
Facets/verf |
|---|---|---|---|---|---|
| 16-cell honeycomb | = [3,3,4,3] | {3,3,4,3} | Template:CDD | Template:CDD [3,4,3], order 1152 |
24: 16-cell |
| 4-demicube honeycomb | = [31,1,3,4] | {31,1,3,4} = h{4,3,3,4} |
Template:CDD = Template:CDD | Template:CDD [3,3,4], order 384 |
16+8: 16-cell |
| = [31,1,1,1] | {31,1,1,1} = h{4,3,31,1} |
Template:CDD = Template:CDD | Template:CDD [31,1,1], order 192 |
8+8+8: 16-cell |
Related honeycombs
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See also
Regular and uniform honeycombs in 4-space:
- Tesseractic honeycomb
- 24-cell honeycomb
- Truncated 24-cell honeycomb
- Snub 24-cell honeycomb
- 5-cell honeycomb
- Truncated 5-cell honeycomb
- Omnitruncated 5-cell honeycomb
Notes
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References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
- pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4} = {4,4}; h{4,3,4} = {31,1,4}, h{4,3,3,4} = {3,3,4,3}, ...
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
- Template:KlitzingPolytopes x3o3o4o3o - hext - O104
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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(I'm no baseball guy, but if you increase the opponent's chance of winning by intentionally putting a runner on first in that situation, how come EVERY manager does it?
The citizens of Summerside and the surrounding area derserved this facility, for too long we had to use outdated decrepit, unsafe and embarassing facilities, I applaud ethe Mayor and the Council at the time who had the vision, and the and the community spirit to push the building of this facility thru all the politics and legal stuff to get this built. And it continues to grow, The addition of a the skateboarding facility tp CUP is proving to be a well used and appreciated park for our young citizens.
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- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html
- ↑ 2.0 2.1 http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Conway (1998), p. 119
- ↑ Conway (1998), p. 120
- ↑ Conway (1998), p. 466